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We are witnessing a rapid transition from Software 1.0 to 2.0. Software 1.0 focuses on manually designed algorithms, while Software 2.0 leverages data and machine learning algorithms (or artificial intelligence) for optimized, fast, and accurate solutions. For the past few years, we have been developing Convolution Hierarchical Deep-learning Neural Network Artificial Intelligence (C-HiDeNN-AI), which enables the realization of Engineering Software 2.0 by opening the next-generation neural network-based computational tools that can simultaneously train data and solve mechanistic equations. This paper focuses on solving partial differential equations with C-HiDeNN. Still, the same neural network can be used for training and calibration with experimental data, which will be discussed in a separate paper. This paper presents a computational framework combining the C-HiDeNN theory with isogeometric analysis (IGA), called Convolution IGA (C-IGA). C-IGA has five key features that advance IGA: (1) arbitrarily high-order smoothness and convergence rates without increasing degrees of freedom; (2) a Kronecker delta property that enables direct imposition of Dirichlet boundary conditions; (3) automatic and flexible global/local mesh-adaptivity with built-in length scale control and adjustable radial basis functions; (4) ability to handle irregular meshes and triangular/tetrahedral elements; and (5) GPU implementation that speeds up the program as fast as finite element method (FEM). Mathematically, we prove that both IGA and C-IGA mappings are equivalent, and by taking a special design and modified anchors as nodes, C-IGA degenerates to IGA. We demonstrate the accuracy, convergence rates, mesh-adaptivity, and performance of C-IGA with several 1D, 2D, and 3D numerical examples. The future applications of C-IGA from topology optimization to product manufacturing with multi-GPU programming are discussed. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Bürchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model's density outperforms that of the velocity — particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) – a combination of isogeometric analysis (IGA) and FCM – is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density – thus allowing to identify voids in a physical object – is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures. [ABSTRACT FROM AUTHOR]

In this paper, a numerical formulation for modeling turbulent multi-phase, multi-fluid flows over real-life marine devices is presented. The flow field is governed by the Navier–Stokes equations with an Arbitrary-Lagrangian–Eulerian (ALE) description of the continuum for handling independent domain movements. The multi-phase phenomenon is modeled for an iso-thermal homogeneous mixture of water vapor and liquid phases using a transport equation for the vapor volume fraction. The multi-fluid phenomenon is modeled using the level-set method where a geometric function, namely, the signed distance function (SDF) describes the level-set field. Turbulence modeling is handled in the context of the variational multiscale (VMS) method resulting in an ALE-VMS formulation for multi-phase, multi-fluid flows over moving hydrodynamic objects. Several augmentations to the formulation are discussed in this paper including the sliding interface (SI) operator and its role in enforcing the compatibility of solution fields between in-contact computational subdomains during the different steps of the solution. This formulation is validated against experimental data for a real-life horizontal-axis marine turbine under different flow conditions including single-phase flows, multi-phase flows and multi-fluid flows. The performance of the turbine as well as the expected flow behaviors are compared to the experimental data with success. An additional case is simulated for a multi-phase, multi-fluid flow condition showing the capabilities and robustness of the presented numerical formulation. • A variational multiscale (VMS) framework for multi-phase, multi-fluid is developed. • Sliding interface formulation is used for components in relative motion. • Simulations of horizontal-axis hydrokinetic turbine at full-scale is performed. • Results are compared to the experimental data showing good agreement. [ABSTRACT FROM AUTHOR]

This paper presents experimental verification of a multi-level simulation framework aimed at sheet metal forming analysis. Specifically, deep drawing of a cylindrical cup simulations from alloy AA6022-T4 sheets are carried out using a physically based elasto-plastic self-consistent (EPSC) polycrystalline homogenization model embedded in implicit finite elements and verified experimentally. The EPSC model takes into account the evolution of microstructure and directionality of deformation mechanisms acting at the single-crystal level in predicting material behavior. It is calibrated and validated as a standalone model using flow stress and R-ratio data as well as iso-shear contours measured along several directions of the sheet through uniaxial and plane strain tension experiments. Furthermore, the particularities pertaining to cyclic response including non-linear unloading and the Bauschinger effect are also calibrated through large strain tension–compression data. Consistent with experimental measurements, the process simulations in finite elements predict directionally dependent thinning of the cup, especially around the punch radius and variation in the cup height around the rim of the cup, referred to as earing. By comparing experiments and predictions, role of the R-ratio is revealed as critical for the accurate prediction of the cup height. Further sensitivity analysis shows that initial texture has a strong influence on the R-ratio, while introducing a minor effect on hardening. The analysis into the choice of finite element types in terms of their accuracy and efficiency shows that the 3D 8 nodal elements (C3D8R) and continuum shell 3D 8 nodal elements (SC8R) are superior over the planar shell 3D 4 nodal elements (S4R) with the former being the most accurate and the latter being computationally efficient. It is demonstrated that the simulation framework presented in this paper can be used to predict phenomena pertaining to material behavior and resulting geometrical changes important for optimization of the sheet metal forming processes. • Grain-to-polycrystal-to-component level models are linked to simulate cup drawing. • Hardening, unloading, Bauschinger effect, and geometrical changes are predicted. • Crystallographic texture is found to govern R-ratio. • Role of R-ratio in predicting cup height is explained. • Continuum 3D elements are recommended for crystal plasticity sheet metal forming. [ABSTRACT FROM AUTHOR]

This study presents a novel multi-scale computational method to analyze the dynamic thermo-mechanical performance of heterogeneous shell structures with orthogonal periodic configurations. The heterogeneities of heterogeneous shell structures are taken into account by periodic layouts of unit cells on the microscale in orthogonal curvilinear coordinate system. The new second-order two-scale approximate solutions for these multi-scale problems are constructed based on the multi-scale asymptotic analysis. Furthermore, the error estimates for the second-order two-scale (SOTS) solutions are obtained under some hypotheses. And then, a novel SOTS numerical algorithm based on finite element method (FEM), finite difference method (FDM) and decoupling method is brought forward in detail. Finally, some numerical examples are presented to verify the feasibility and validity of our multi-scale computational method. They also demonstrate that our multi-scale computational method can accurately capture the micro-scale dynamic thermo-mechanical responses in heterogeneous block structure, plate, cylindrical and doubly-curved shallow shells. In this paper, a unified multi-scale computational framework is established for dynamic thermo-mechanical problems of heterogeneous materials and structures with orthogonal periodic configurations. The asymptotic homogenization theory in Cartesian coordinate system and cylindrical coordinate system can be directly obtained based on the results in this paper. • A multi-scale computational method is built for dynamic thermo-mechanical problems of heterogeneous shell structures. • The novel multi-scale method is effective and stable after long-time numerical simulation. • The convergence result with an explicit rate for the SOTS solutions is obtained in the integral sense. • The asymptotic homogenization theory in cartesian coordinates is extended into orthogonal curvilinear coordinates. [ABSTRACT FROM AUTHOR]

Shape memory materials have gained considerable attention thanks to their ability to change physical properties when subjected to external stimuli such as temperature, pH, humidity, electromagnetic fields, etc. These materials are increasingly used for a large number of biomedical applications. For applications inside the human body, contactless control can be achieved by the addition of electric and/or magnetic particles that can react to electromagnetic fields, thus leading to a composite biomaterial. The difficulty of developing accurate numerical models for smart materials results from their multiscale nature and from the multiphysics coupling of involved phenomena. This coupling involves electromagnetic, thermal and mechanical problems. This paper contributes to the multiphysics modeling of a shape memory polymer material used as a medical stent. The stent is excited by electromagnetic fields produced by a coil which can be wrapped around a failing organ. In this paper we develop large deformation formulations for the coupled electro-thermo-mechanical problem using the electric potential to solve the electric problem. The formulations are then discretized and solved using the finite element method. Results are validated by comparison with results in the literature. [ABSTRACT FROM AUTHOR]

This paper presents an efficient solution for designing the topology of structures that can withstand dynamic loads. The method, called image/stereolithography (STL)-based adaptive scaled boundary finite element (SBFE), is a novel approach to topology optimization (TO) that is particularly effective for designing two/three-dimensional structures. The SBFE-based TO algorithm adopts a bi-evolutionary structural optimization (BESO) approach, which utilizes image/STL-based procedures to automatically adapt meshes and minimize mechanical compliance under varying volume fractions in the time domain. A key advantage of this approach is that it uses images of TO responses, rather than specific error functions, to guide the automatic adaptive mesh schemes. The method also incorporates a quadtree/octree mesh refinement to address the problem of hanging nodes, ensuring fast mesh convergence during the optimization process. Furthermore, a high-order implicit time integration technique provides a close approximation of equivalent static loads, allowing for the accurate mapping of displacement fields under actual dynamic responses at each time step. The effectiveness of this method is demonstrated through a variety of numerical examples and benchmarks. • SBFE-based TO algorithm adopts image/STL procedures for automated mesh adaptation. • Multiple dynamic loadings applied in optimization result in various optimal layouts. • Implicit time integration based on Padé expansions is employed in elastodynamic analysis. • Objective function, weighted summation of compliances near critical peak points, reduces CPU time. • Optimal results undergo further smoothing using Fusion 360, a commercial software. [ABSTRACT FROM AUTHOR]

Accurate material description is crucial to achieve high-quality results in computational analysis software. Phenomenological constitutive laws generalize the material behaviour observed in simple mechanical tests. The resulting empirical expressions contain parameters that need to be calibrated through an inverse optimization process. Advancements in Digital Image Correlation (DIC) techniques have enabled the extraction of non-uniform multi-axial displacement fields, facilitating the development of heterogeneous mechanical specimens and unlocking access to richer material behaviour data. Despite these advancements, constitutive models are still limited by mathematical formulations and biases due to simplifying assumptions. Artificial Neural Networks (ANNs), as universal function approximators, could potentially drive a paradigm shift in the field. ANNs do not require explicit pre-formulations, hence avoiding the need for identifying unknown parameters. Moreover, ANNs are able to implicitly learn patterns purely from data, allowing to circumvent the biases induced by the simplifying assumptions of first-order principles. However, the training of ANNs for implicit constitutive modelling is not straightforward, given the impossibility of obtaining direct measurements of stress. This paper explores the integration of Recurrent Neural Networks (RNNs) with the Virtual Fields Method (VFM) for material modelling. This approach uses global force and displacement data to indirectly train the neural network, with the equilibrium being evaluated globally through the VFM loss function. The proposed method is (i) computationally efficient by not requiring Finite Element Analysis (FEA), (ii) compatible with both full-field measurements and numerically generated data, and (iii) able to handle experimental boundary conditions. • A novel method integrates RNNs with the VFM for implicit material modelling. • The RNN is trained without labels, using only displacement/strain and global force. • The approach is real-test oriented, compatible with both DIC and numerical data. • The methodology is independent of the boundary conditions. • The method is computationally more efficient than FEA indirect training. [ABSTRACT FROM AUTHOR]

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realize the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues. [ABSTRACT FROM AUTHOR]

Despite substantial advancements in optimizing structural designs for stiffness, the field of design against fracture is still in its early stages. This paper introduces a fundamental and standardized paradigm for structural design against brittle fracture, achieved through the minimization of the energy release rate G within the framework of linear elastic fracture mechanics (LEFM). By leveraging the Griffith criterion expression and the total potential energy in LEFM, we propose an efficient approach for structural optimization that enhances the toughness of architected materials. The energy release rate G is evaluated a synergistic application of finite element analysis and finite difference methods, yielding a precision of 0.6% compared to benchmarks in published handbooks. We demonstrate significantly reduced G values compared to standard stiffness and stress-oriented topology optimization, which directly contributes to increased toughness. To comprehensively analyze the fracture process encompassing damage, crack initiation, propagation, and failure, we employ post-processing phase field fracture modeling on the optimally designed architected materials. By performing multi-objective optimization of the stiffness and energy release rate G , the toughness and extreme load-bearing capacity can be simultaneously improved. The resulting perforated architecture effectively outperforms homogeneous structures in both toughness and peak load. Additionally, we extend our analysis to various configurations, such as non-center cracks, pure shear loading, and adherence to the standard 3D compact tension ASTM E-399-72. The designed structures are additively manufactured and experimentally validated, demonstrating a toughness increase of over 10 times compared to stress-based design. These contributions offer promising prospects for advancing the design of future engineering structures and materials with a favorable balance of critical fracture resistance characteristics. [ABSTRACT FROM AUTHOR]

This paper presents an extension of the discrete element method using a phase-field formulation to incorporate grain shape and its evolution. The introduction of a phase variable enables an effective representation of grain geometry and facilitates the application of physical laws, such as chemo-mechanical couplings, for modeling shape changes. These physical laws are solved numerically using the finite element method coupled in a staggered scheme to the discrete element model. The efficacy of the proposed Phase-Field Discrete Element Model (PFDEM) is demonstrated through its ability to accurately capture the real grain shape in a material subjected to dissolution only and compute the stress evolution. It is then applied to model the phenomenon of pressure solution involving dissolution and precipitation in granular materials at the microscale and enables to reproduce the creep response observed experimentally. This framework contributes to the enhanced understanding and simulation of complex behaviors in granular materials and sedimentary rocks for many geological processes like diagenesis or earthquake nucleation. • Coupling between PF and DEM to incorporate grain shape and its evolution. • The grain shape influes the behavior to an oedometer test under acid injection. • Model the phenomenon of pressure solution at the microstructural level. [ABSTRACT FROM AUTHOR]

The paper deals with the rebound of an elastic solid off a rigid wall of a container filled with an incompressible Newtonian fluid. Our study focuses on a collision-free bounce, meaning a rebound without topological contact between the elastic solid and the wall. This has the advantage of omitting any artificial bouncing law. In order to capture the contact-free rebound for very small viscosities an adaptive numerical scheme is introduced. The here-introduced scheme is based on a Glowinski time scheme and a localized arbitrary Lagrangian–Eulerian map on finite elements in space. The absence of topological contact requires that very thin liquid channels are solved with sufficient accuracy. It is achieved via newly developed geometrically driven adaptive strategies. Using the numerical scheme, we present here a collection of numerical experiments. A rebound is simulated in the absence of topological contacts. Its physical relevance is demonstrated as, with decreasing viscosities, a free rebound in a vacuum is approached. Further, we compare the dynamics with a second numerical scheme; a here-introduced adaptive purely Eulerian level-set method. The scheme produced the same dynamics for large viscosities. However, as it requires a much higher computational cost, small viscosities cannot be reached by this method. The experiments allow for a better understanding of the effect of fluids on the dynamics of elastic objects. Several observations are discussed, such as the amount of elastic and/or kinetic energy loss or the precise connection between the fluid pressure and the rebound of the solid. [Display omitted] • Contactless rebound of elastic solids in fluids. • Viscosity convergence of (contactless) rebound to vacuum case. • Updated arbitrary Lagrangian–Eulerian method with re-meshing. • A new re-meshing strategy based on the solution of the Eikonal equation. [ABSTRACT FROM AUTHOR]

The computational analysis of nanophotonic devices is usually carried out via the standard finite element method (FEM). However, FEM requires meshes that are fitted to the devices' boundaries, so making changes to the geometry (and thus the mesh) results in an inefficient process at best. Such an approach is therefore at odds when conducting design, which requires the analysis of multiple device geometries until reaching a satisfactory solution. Computational design tools such as topology optimization are often used, but the use of density-based representations of geometry inevitably leads to other issues—e.g., pixelized fuzzy boundaries with "gray material" (that does not correspond to dielectric nor vacuum) have an adverse effect on the devices' interaction with electromagnetic waves. In this paper we propose an interface-enriched generalized finite element method (IGFEM) for the analysis of two-dimensional electromagnetic scattering and eigenvalue problems. IGFEM enables the use of finite element meshes that are completely decoupled from the problem's geometry. The analysis procedure is further coupled to a level set description of topology, resulting in a versatile enriched approach to topology optimization; this level set-based interface-enriched topology optimization procedure is devoid of the issues mentioned above regarding density-based methods, and yields crisp "black-and-white" designs that are devoid of jagged fuzzy edges. We first demonstrate that the analysis procedure achieves the same convergence rate as that of standard FEM using geometry-fitted meshes. We then compare the convergence properties of IGFEM with Nitsche's method on a problem containing an embedded straight interface. Finally, we conduct topology optimization for designing both a 2-D metalens and a 2-D reflector, maximizing their ability to focus light onto a target point. [Display omitted] • Interface-enriched GFEM with nodal elements allows to decouple mesh from geometry. • Nodal enrichments allow for straightforward implementation in displacement-based FEM codes. • IGFEM-based analysis has optimal error convergence in L 2 norm for smooth problems. • Photonic crystal band structures can be computed accurately with IGFEM-based analysis. • Enriched level set-based topology optimization yields black-and-white smooth designs. [ABSTRACT FROM AUTHOR]

Abstract The static solution of crack propagation problems can be an efficient way to find both the failure load of a structure and the shape of the crack pattern. The paper introduces two new implicit static solution procedures to study crack propagation problems by adopting a Peridynamic-based numerical tool, and compares them with the Sequentially Linear Analysis. We discretize the structures in space by adopting a coupled FEM–PD approach, which exploits the flexibility of FEM to reduce the overall computational cost of the simulation. The results of several numerical examples indicate the main novel conclusions of the paper: when using a PD-based software, controlling the maximum number of bonds broken in each iteration may increase significantly the accuracy of the solution and keep the computational cost to an acceptable level. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of "subtracting the interpolant." A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface. Highlights • GFEM for 2D non-smooth interface problem. • Singular enrichment functions in addition to distance based enrichment functions. • Proof of optimal convergence of the GFEM. • Experimental study of conditioning and robustness of GFEM. • The notion of Strongly Stable GFEM (SSGFEM). [ABSTRACT FROM AUTHOR]

Abstract We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Such multimesh finite element methods are particularly well suited to problems in which the computational domain undergoes large deformations as a result of the relative motion of the separate components of a multi-body system. In the present paper, we formulate the multimesh finite element method for the Poisson equation. Numerical examples demonstrate the optimal order convergence, the numerical robustness of the formulation and implementation in the face of thin intersections and rounding errors, as well as the applicability of the methodology. In the accompanying paper (Johansson et al., 2018), we analyze the proposed method and prove optimal order convergence and stability. [ABSTRACT FROM AUTHOR]

Modeling deformation processes of materials under high strain rate and impact conditions in which the strain rates vary spatiotemporally over several orders of magnitude is challenging, especially in terms of constitutive description. Visco-plastic power-law flow rule, commonly used within crystal plasticity constitutive models, usually introduces superfluous strain rate effects, entering numerically via the slip activation criterion, which result in artificially high flow stress predictions under high strain rate deformation conditions. This paper presents a novel, implicit finite element implementation of a visco-plastic self-consistent polycrystal model that is implicit in dislocation-based hardening with removed strain rate sensitivity introduced by the power-law flow rule. In the model, the strain rate dependence of the flow stress is defined solely by the rate sensitivity of Peierls stress and the evolution of dislocation density within the hardening law, enabling the prediction of high strain rate and impact deformation. As a result, the predicted flow stress is accurate in its magnitude. The physically based strain rate and temperature sensitive model also features, in addition to Schmid, the non-Schmid activation contribution for slip, which is necessary for understanding and modeling the plastic deformation of body-centered cubic metals. The model is calibrated to simulate the strain rate- and temperature-sensitive monotonic deformation of tantalum and is subsequently applied to a Taylor impact test of the same material. Since the impact simulation required remeshing and interpolation of state variables, the spherical linear interpolation algorithm in the space of quaternions is implemented to facilitate the interpolation of texture. Predictions of the simulation were found to compare favorably with experimental measurements of post-test geometrical changes and texture evolution. The implementation and insights from these predictions are presented and discussed in the paper. [ABSTRACT FROM AUTHOR]

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier–Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective–diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective–diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259–280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection–diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov–Galerkin (SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method. [ABSTRACT FROM AUTHOR]

For integrals with various orders of vertex singularity encountered in the generalized/extended finite element method, three difficulties should be concerned as: (a) the singularity with respect to local distance r ; (b) the near singularity caused by ill-shaped patches; (c) the near singularity during the iso-parametric transformation. In this paper, based on the Duffy transformation, the near singularities caused by ill-shaped patches and element shape distortion are revealed theoretically and numerically, a Duffy-distance transformation is constructed to damp out the near singularity caused by ill-shaped patches, and a conformal preconditioning strategy is developed to fully eliminate the near singularity caused by element shape distortion. Extensive numerical examples, including crack-tip elements with distorted shape, are given to validate the efficiency and robustness of the presented scheme in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with the implementation of thermomechanically coupled gradient-enhanced elastoplasticity at finite strains. The presented algorithmic formulation heavily relies on the variational structure of the considered initial boundary value problem. Consequently, such a variational structure is elaborated. While variational formulations are well-known in the case of isothermal plasticity theory, thermomechanically coupled gradient-enhanced plasticity theory has not been considered before. The resulting time-continuous variational principle allows the computation of all unknown variables jointly and naturally from the stationarity condition of an incremental potential. By discretization of this time-continuous potential in time, a discrete approximation is obtained which represents the foundation of the algorithmic formulation. As a matter of fact, stationarity of the respective potential again defines all unknown variables — now in a time-discrete fashion. Within this paper, the necessary condition associated with stationarity – a vanishing first gradient – is solved numerically by means of Newton’s method. Hence, the first as well as the second derivatives of the incremental potential are required. They are computed by numerical differentiation based on hyper-dual numbers. By considering a perturbation with respect to hyper-dual numbers, the first as well as the second derivatives are computed in an exact manner without introducing any numerical errors. Although numerical differentiation based on hyper-dual numbers is numerically more extensive than real-valued perturbations, it is shown that the scalability of the resulting parallel finite element implementation is not dominated by this effect for sufficiently large problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the contrast and linearly decreases with respect to mesh size if oversampling size is appropriately chosen. We would like to show a mesh-dependent convergence with a minimal number of basis functions. Our construction starts with an auxiliary multiscale space by solving local spectral problems. In auxiliary multiscale space, we select the basis functions that correspond to small (contrast-dependent) eigenvalues. These basis functions represent the channels (high-contrast features that connect the boundaries of the coarse block). Using the auxiliary space, we propose a constraint energy minimization to construct multiscale spaces. The minimization is performed in the oversampling domain, which is larger than the target coarse block. The constraints allow handling non-decaying components of the local minimizers. If the auxiliary space is correctly chosen, we show that the convergence rate is independent of the contrast (because the basis representing the channels are included in the auxiliary space) and is proportional to the coarse-mesh size (because the constraints handle non-decaying components of the local minimizers). The oversampling size weakly depends on the contrast as our analysis shows. The convergence theorem requires that channels are not aligned with the coarse edges, which hold in many applications, where the channels are oblique with respect to the coarse-mesh geometry. The numerical results confirm our theoretical results. In particular, we show that if the oversampling domain size is not sufficiently large, the errors are large. To remove the contrast-dependence of the oversampling size, we propose a modified construction for basis functions and present numerical results and the analysis. [ABSTRACT FROM AUTHOR]

Interpolation of motion is required in various fields of engineering such as computer animation and vision, trajectory planning for robotics, optimal control of dynamical systems, or finite element analysis. While interpolation techniques in the Euclidean space are well established, general approaches to interpolation on manifolds remain elusive. Interpolation schemes in the Euclidean space can be recast as minimization problems for weighted distance metrics. This observation allows the straightforward generalization of interpolation in the Euclidean space to interpolation on manifolds, provided that a metric of the manifold is defined. This paper proposes four metrics of the motion manifold: the matrix, quaternion, vector, and geodesic metrics. For each of these metrics, the corresponding interpolation schemes are derived and their advantages and drawbacks are discussed. It is shown that many existing interpolation schemes for rotation and motion can be derived from the minimization framework proposed here. The problems of averaging of rotation and motion can be treated easily within the same framework. Both local and global interpolation problems are addressed. The proposed interpolation framework can be used with any suitable set of basis functions. Examples are presented with Chebyshev spectral, Fourier spectral, and B-spline basis functions. This paper also introduces one additional approach to the interpolation of motion based on the interpolation of its derivatives. While this approach provides high accuracy, the associated computational cost is high and the approach cannot be used in multi-variable interpolation easily. [ABSTRACT FROM AUTHOR]

The local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high-order finite elements in the 1-D, 2-D and 3-D cases on uniform and non-uniform meshes. Several applications of the local truncation error are considered in the paper. It is proven that for the 1-D wave equation with a piece-wise constant wave velocity, the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity, the lumped mass matrix and the central-difference method with the time increments equal to the stability limit are used. It is shown in the 1-D and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy. The minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high-order elements with improved accuracy; i.e., accuracy is improved from order 2 p (the conventional elements) to order 4 p (the new elements) where p is the order of the polynomial approximations. New high-order boundary conditions are developed in order to keep a high-order accuracy of the developed technique. The new elements can be equally applied to linear wave propagation and heat transfer problems. It is also shown that non-uniform meshes may lead to inaccurate results due to the increase in the local truncation error. The difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach. The presented numerical examples are in good agreement with the theoretical results. The approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches. [ABSTRACT FROM AUTHOR]

• The added degree of freedom method (ADM) is proposed to solve the absorbing boundary problem. • The principle of a mass damper is employed in the ADM to achieve the attenuation of stress waves. • The ADM can absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Absorbing boundaries are essential in engineering simulations, especially for elastodynamic problems, to ensure accuracy, reliability and numerical stability. In this paper, the added degree of freedom method (ADM) is developed as a novel approach to address absorbing boundary challenges in finite element analysis. In ADM, additional degrees of freedom (DOFs) are introduced within the absorbing domain to attenuate outgoing elastic waves. For the proposed ADM to implement the absorbing boundary, the stiffness and mass properties of both the added DOFs and conventional DOFs within the absorbing domain are adjusted. This adjustment aims to reduce the propagation speed of elastic waves within the medium, and to prolong the duration of interaction between elastic waves and the surrounding medium. Consequently, the vibrations of nodes can be effectively attenuated by applying damping forces to the added DOFs. The numerical results show that the ADM has the ability to absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Therefore, the proposed ADM offers an innovative solution for modeling absorbing boundaries in various scientific and engineering applications, addressing the challenges of simulating wave propagation within finite computational domains. [ABSTRACT FROM AUTHOR]

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces. [ABSTRACT FROM AUTHOR]

The Scaled Boundary Finite Element Method (SBFEM) can be viewed as a semi-analytical approach, which has been successfully applied to singular mechanical problems with vanishing force terms. Interpreted in the context of composite Duffy approximations based on star-shape polytopes (S -elements), and using an operator-adapted principle, the shape functions are determined by imposing an energy-orthogonality constraint. In the presence of non-vanishing forcing terms, the SBFEM approximation converges at sub-optimal rates. In the current paper, a strategy is proposed to overcome this drawback by the enrichment of the classic SBFEM spaces with energy-orthogonal bubble Duffy functions. The orthogonality property leads to a decoupling of the approximate solutions into components of boundary trace and interior (bubble) functions. The bubble functions can be solved separately, thus reducing the computational cost. A priori energy-error estimates are provided demonstrating optimal rates of convergence. The convergence rates are confirmed by two and three-dimensional numerical verification tests. • Operator-adapted SBFEM approximations in the presence of body force are proposed. • A priori error estimates for SBFEM with bubble enrichment are provided. • Optimal convergence rates for Poisson and elasticity problems are numerically verified. • The SBFEM bubble-enriched approximations hold energy-orthogonal properties. • The orthogonality property leads to a reduction in the computational cost. [ABSTRACT FROM AUTHOR]

Numerical prediction of melt pool morphology and temperature distribution of thermomechanical processes (welding, additive manufacturing) plays an important role in understanding the relationships between process parameters and the quality of manufactured parts. The heat conduction models are limited in their predictability because the transport phenomena relevant to the melt pool dynamic are ignored. In contrast, the multiphysics model allows consideration of the heat transfer, free surface evolution, and transport phenomena in the melt pool, leading to more accurate predictions. However, the computational cost of multiphysics models makes them impractical for part-scale simulations. In this paper, a local moving thermal-fluid framework based on the finite element method is presented, which allows solving the thermal-fluid problem only in the small moving zone containing the melt pool and the heat transfer problem in the rest part. Therefore, this new proposed moving thermal-fluid (MTF) framework is predictive as well as the full thermal-fluid model, while it is much more efficient than a full thermal-fluid model since there are much fewer degrees of freedom (DOF) to solve. Finally, numerical validation tests in welding and additive manufacturing simulations highlight its computational efficiency and high-fidelity, and a relative error of less than 2.6% was observed in predicting melting pool dimensions compared to full thermal-fluid model. In the part-scale simulation of laser welding benchmark, the MTF model takes only 62 h in place of about three weeks required by a full thermal-fluid model. • A computational efficient local moving thermal-fluid framework is proposed. • The thermal-fluid problem is solved only in the small zone containing the melt pool. • The new model is applied for modeling welding and additive manufacturing processes. • The accuracy and efficiency of new approach is verified against numerical references. [ABSTRACT FROM AUTHOR]

This paper presents an approach to the numerical estimation of effective properties of highly porous materials based on the scaled boundary finite element method (SBFEM). The latter can be formulated on quadtree meshes with hanging nodes and thus facilitates the efficient mesh generation and analysis of a large number of randomly created samples. To generate the corresponding Representative Volume Elements (RVEs), an improved Random Sequential Addition (RSA) method with overlap is used. Moreover, an alternative image-based digital matrix method for generating random assemblies of circular particles of the same size is proposed. In addition, a modified smoothing algorithm to treat the staircase boundaries present in a quadtree mesh is developed in order to accurately and efficiently capture the sinter-neck features of overlapping circular particles. The proposed image-based mesh generation and analysis procedure is used to evaluate the relationship between microstructural parameters and effective Young's modulus considering a large number of samples. In this context, a new dimensionless morphological parameter N f c is proposed. The latter represents the morphology and connectivity of voids and is shown to be a useful indicator with respect to the effective elastic properties of highly porous ceramic matrix material. • Scaled boundary finite element method for homogenization analysis of highly porous materials. • Model generation method for large number of Representative Volume Elements (RVE) of porous ceramic matrix material. • New automatic quadtree meshing and smoothing algorithm with high accuracy and efficiency for complex structures. • New dimensionless parameter that exhibits a stronger correlation with the material's geometric properties and effective elastic properties than porosity. [ABSTRACT FROM AUTHOR]

This paper introduces an extended tensor decomposition (XTD) method for model reduction. The proposed method is based on a sparse non-separated enrichment to the conventional tensor decomposition, which is expected to improve the approximation accuracy and the reducibility (compressibility) in highly nonlinear and singular cases. The proposed XTD method can be a powerful tool for solving nonlinear space–time parametric problems. The method has been successfully applied to parametric elastic–plastic problems and real time additive manufacturing residual stress predictions with uncertainty quantification. Furthermore, a combined XTD-SCA (self-consistent clustering analysis) strategy is presented for multi-scale material modeling, which enables real time multi-scale multi-parametric simulations. The efficiency of the method is demonstrated with comparison to finite element analysis. The proposed method enables a novel framework for fast manufacturing and material design with uncertainties. • Novel eXtended Tensor Decomposition (XTD) based nonlinear model reduction method. • Sparse non-separated enrichment can improve the accuracy, convergence, and robustness of model reduction methods. • XTD has been applied to nonlinear elastic–plastic problems for stress predictions in additive manufacturing. • The combined XTD-SCA (self-consistent clustering analysis) can enable fast multiscale material modeling and design under uncertainty. [ABSTRACT FROM AUTHOR]

This paper develops an adaptive algorithm for the Generalized Finite Element Method with global–local enrichment (GFEM gl) for transient multiscale PDEs. The adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, at each time step, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEM gl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values. The proposed approach is shown to be robust for solving advection–diffusion problems that require detecting initial conditions with spikes and capturing moving/morphing/merging mass plumes in heterogeneous media and non-uniform flow with sharp fronts. Specific examples are provided for applications in groundwater contaminant and heat dissipation. The accuracy of the proposed adaptive GFEM gl closely matches reference fine-scale FEM solutions in the L ∞ norm. [ABSTRACT FROM AUTHOR]

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples. • An efficient concurrent topology optimization method for designing hierarchical structures with multiple substructures is proposed based on the Extended Multiscale Finite Element Method (EMsFEM). • Without aforehand limitation on substructures' topologies and their spatial distribution, the proposed method can simultaneously optimize the topologies and spatial distribution of substructures in hierarchical structures. • The proposed method is free of the assumption of separation of length scales, which not only avoids the connection problem between adjacent substructures but also makes the manufacturing of optimized hierarchical structures of practical significance. [ABSTRACT FROM AUTHOR]

In this paper, an efficient formulation for the topology optimization of extruded beams is developed considering linearized buckling, displacement, and stress constraints. The proposed method relies on a computationally efficient extended\generalized finite element (XFEM\GFEM) formulation for the forward problem, recently developed for the analysis of beams. In this method, a coarse 3D finite element (FE) mesh is enhanced with global enrichment functions, enabling a substantial reduction in the number of degrees of freedom with comparable accuracy to a traditional 3D FE fine mesh. The material distribution at the cross-section is optimized utilizing the material density approach with the Solid Isotropic Material with Penalization (SIMP) method. Our objective is to minimize the total weight of the beam, subject to performance constraints including buckling, displacement, and stress. The forward problem enables the application of general loading and boundary conditions. Thus, both buckling and lateral–torsional buckling are automatically considered. The optimization problem is solved using a gradient-based optimization approach, with analytical gradients derived by using the adjoint variable method. Several numerical examples are presented to investigate the accuracy and efficiency of the proposed formulation. A parametric study showed that the optimized cross-section is strongly dependent on the length and permissible buckling load. In addition, it is shown that a simultaneous formulation leads to significantly different designs, with different structural behavior, compared to results obtained by considering only a subset of the performance constraints. Finally, we verify the model on a beam warping problem under pure torsion, in which the cross-section optimization converges to a known analytical solution. Compared with traditional FEM, two orders of magnitude in terms of execution time are reported in favor of the proposed XFEM\GFEM approach. • A new approach for the topology optimization of extruded beams is proposed. • The forward problem is solved using an efficient numerical scheme relying on the extended finite element method. • The approach simultaneously considers buckling, stress, and displacement constraints. • We show that the beam's length significantly affects the optimal shape. • The proposed XFEM\GFEM approach shows significant time savings compared with 3D FE. [ABSTRACT FROM AUTHOR]

This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high-order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. [ABSTRACT FROM AUTHOR]

In this paper, a peridynamic-enhanced finite element formulation is introduced for the numerical simulation of thermo–hydro–mechanical coupled problems in saturated porous media with cracks. The proposed approach combines the Finite Element (FE) method for governing heat conduction–advection and fluid flow in the fractured porous domain, and the Peridynamic (PD) method for describing solid phase deformation and capturing crack propagation. Firstly, the consolidation problem of a porous column is simulated by using the proposed approach. The m - and δ -convergence studies are carried out with the isothermal condition to determine suitable discretization parameters for the PD model. Subsequently, non-isothermal conditions are considered, and the accuracy and reliability of the proposed approach are validated by comparing the numerical solutions with those obtained from a FE-only model. Furthermore, several numerical examples focusing on scenarios involving cracks are solved and presented to further highlight the capabilities of the proposed approach in addressing heat conduction–advection problems in fractured saturated porous media, as well as hydraulic fracture propagation problems with considerations of thermo–hydro–mechanical coupled effects. • A PD-FE model is used for THM-coupled problems in saturated fractured porous media. • Detailed nonlocal coupling matrices for hydraulic and thermal forces are provided. • The staggered solution algorithm is introduced to solve the THM-coupled system. • Several numerical cases are presented to show the capabilities of the proposed model. [ABSTRACT FROM AUTHOR]

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning, including Gaussian process regression and statistical finite element analysis. Today's prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest in the statistics and engineering literature. The SPDE representation is especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on geometrically complex domains. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix do not contain dense matrices and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples. • A statistical approach for synthesising large-scale FE models and data is proposed. • Gaussian random fields are expressed as the solution of fractional stochastic PDEs. • As observation models Gaussian process regression and statFEM are considered. • Posterior mean and covariance are determined using only sparse matrix operations. • Applications to large-scale Poisson and thin-shell problems are demonstrated. [ABSTRACT FROM AUTHOR]

This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework's performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem). [ABSTRACT FROM AUTHOR]

In this paper, a novel higher stabilization-free virtual element method is proposed for compressible hyper-elastic materials in 2D. Different from the most traditional virtual element formulation, the method does not need any stabilization. The main idea is to modify the virtual element space to allow the computation of a higher-order polynomial L 2 projection of the gradient. Based on that the stiffness matrix can be obtained directly which greatly simplifies the analysis process, especially for nonlinear problems. Hyper-elastic materials are considered and some benchmark nonlinear problems are solved to verify the capability and accuracy of the stabilization-free virtual element method. [ABSTRACT FROM AUTHOR]

The paper develops a finite element method for the Navier–Stokes equations of incompressible viscous fluid in a time-dependent domain. The method builds on a quasi-Lagrangian formulation of the problem. The paper provides stability and convergence analysis of the fully discrete (finite-difference in time and finite-element in space) method. The analysis does not assume any CFL time-step restriction, it rather needs mild conditions of the form Δ t ≤ C , where C depends only on problem data, and h 2 m u + 2 ≤ c Δ t , m u is polynomial degree of velocity finite element space. Both conditions result from a numerical treatment of practically important non-homogeneous boundary conditions. The theoretically predicted convergence rate is confirmed by a set of numerical experiments. Further we apply the method to simulate a flow in a simplified model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced computed tomography images. [ABSTRACT FROM AUTHOR]

In this paper we present a new continuous/discontinuous finite element method with continuous trace space for Stokes problem. This new method has two new bi-linear forms which are responsible for the properties of the method. One of these two new bi-linear forms allows to eliminate completely the discontinuous pressure at element level preserving the convergence rates. The other bi-linear form allows interpolation of any order for pressure to be possible since the order of interpolation for the velocity and for the trace variable or hybrid variable satisfies a limit well determined. The I n f S u p stability condition is well established for any order of interpolation of the pressure and for arbitrary meshes. Numerical experiments indicate optimal rates of convergence, and that, for the Stokes problem, the method presented in this paper has computational cost lower than that of the Galerkin method with continuous or discontinuous pressure and of the HDG method with discontinuous trace space, for solving the global system using direct solver. Finally, it must be observed that the formulation and the analysis presented here are naturally extended if we used discontinuous trace space. [ABSTRACT FROM AUTHOR]

This paper presents a high order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields analyses and its particularisation to the case of small strains through a staggered scheme. With an emphasis on accurate geometrical representation, a high performance curvilinear finite element framework based on an a posteriori mesh deformation technique is developed to accurately discretise the underlying displacement-potential variational formulation. The performance of the method under near incompressibility and bending actuation scenarios is analysed with extremely thin and highly stretched components and compared to the performance of mixed variational principles recently reported by Gil and Ortigosa (2016) and Ortigosa and Gil (2016). Although convex multi-variable constitutive models are elliptic hence, materially stable for the entire range of deformations and electric fields, other forms of physical instabilities are not precluded in these models. In particular, physical instabilities present in dielectric elastomers such as pull-in instability, snap-through and the formation, propagation and nucleation of wrinkles and folds are numerically studied with a detailed precision in this paper, verifying experimental findings. For the case of small strains, the essence of the approach taken lies in guaranteeing the objectivity of the resulting work conjugates, by starting from the underlying convex multi-variable internal energy, whence avoiding the need for further symmetrisation of the resulting Maxwell and Minkowski-type stresses at small strain regime. In this context, the nonlinearity with respect to electrostatic counterparts such as electric displacements is still retained, hence resulting in a formulation similar but more competitive with the existing linearised electro-elasticity approaches. Virtual prototyping of many application-oriented dielectric elastomers are carried out with an eye on pattern forming in soft robotics and other potential medical applications. [ABSTRACT FROM AUTHOR]

In this paper we study the adaptive finite element method for parabolic equations with Dirac measure. Two kinds of problems with separate measure data in time and measure data in space are considered. It is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the Dirac measure, and thus fit to adaptive FEM for space discretization and variable time steps for time discretization. For both cases we use piecewise linear and continuous finite elements for the space discretization and backward Euler scheme, or equivalently piecewise constant discontinuous Galerkin method, for the time discretization, the a posteriori error estimates based on energy and L 2 norms for the fully discrete problems are then derived to guide the adaptive procedure. Numerical results are provided at the end of the paper to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we present the theoretical justification for the Stable Generalized Finite Element Method (SGFEM) when applied to smooth interface problems. We prove that a Generalized Finite Element Method (GFEM) is stable , i.e., its conditioning is(a) not worse than that of the Finite Element Method, and (b) robust with respect to the mesh, if the enrichment space of the GFEM satisfies two axioms. We provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied. We show that the enrichment space of the GFEM, used to address the interface problems in 2D, satisfies the sufficient conditions and thus the two axioms yielding an SGFEM. The idea of a strongly stable GFEM associated with one of the two axioms has been introduced; strong stability of the GFEM is important for designing efficient iterative methods to solve the underlying linear system. A proof of the optimal convergence of the GFEM for the interface problem has also been derived in this paper. The work in this paper is the continuation of Kergrene et al. (2016), where the stability of the GFEM, when applied to interface problems, was established through numerical experiments. The numerical results in Kergrene et al. (2016) indicated that the GFEM is indeed strongly stable. [ABSTRACT FROM AUTHOR]

The present paper focuses on the discretization and implementation of the Thick Level Set (TLS) approach for quasi-brittle failure when dealing with both local and non-local development of damage. The TLS damage model introduces a length scale and nonlocality in the Fracture Process Zone (FPZ) and it handles the continuous to discontinuous failure transition. The added nonlocality is limited to the FPZ and therefore the model is local away from the FPZ or before its emergence. It leads to concurrent development of local and nonlocal damage during simulations. This paper presents a space–time discretization that handles concurrent computations in a unified manner. It is based on an explicit time discretization written as a predictor–corrector scheme and a space discretization that uses specific approximation functions (modes) that embed nonlocality. It also uses an alternative implementation of the eXtended Finite Element Method (X-FEM) enrichment used in the TLS which is based on virtual nodes. [ABSTRACT FROM AUTHOR]

This paper contributes to the improvement of recently published higher-order accurate energy–momentum schemes for an anisotropic hyperelastic material formulation based on the concept of structural tensors. These Galerkin-based energy–momentum schemes are developed by means of time finite elements with the focus on improving numerical stability and robustness in the presence of stiffness and large rotations. They show advantages over conventional time stepping schemes in combination with anisotropic materials formulated with polyconvex strain energy density functions. In this paper, higher-order energy–momentum schemes are developed by using a differential variational principle in order to remedy the drawbacks of these Galerkin-based schemes, namely a variationally inconsistent time approximation of the strains in the higher-order case, no coordinate free superimposed stress tensor and an unseparated time approximation of the fibers and the matrix material. These new structure-preserving time integrators preserve all conservation laws of a free flying hyperelastic anisotropic continuum, namely the total linear and the total angular momentum conservation as well as the total energy conservation. Numerical examples demonstrate dynamic behavior of anisotropic continua, discrete conservation properties of flying continua and the effect of higher-order accuracy subject to transient loads. [ABSTRACT FROM AUTHOR]

The present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D 3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D 3 symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D 2 , i.e. orthotropic, square unit cell and nonabelian D 3 , i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D 3 symmetric hexagonal unit cell. [ABSTRACT FROM AUTHOR]

A well-known problem in the field of isogeometric analysis is that of surface-to-volume parameterization. In CAD packages, solid objects are represented by a collection of NURBS or T-spline surfaces, but to perform engineering analysis for many real world problems, we must find a way to parameterize the volumes of these objects as well. This has proven to be difficult using traditional isogeometric methods, as the tensor-product nature of trivariate NURBS and T-splines limit their ability to provide analysis suitable parameterizations of arbitrarily complex volumes. To overcome the limitations of trivariate NURBS and T-splines, we propose the use of unstructured Bernstein–Bézier discretizations. In earlier work, we demonstrated the feasibility of this approach in two dimensions through the construction of an automatic mesh generation environment capable of generating geometrically exact unstructured meshes comprised of rational Bézier triangles. This paper extends the concepts presented in our previous work to unstructured tetrahedral and mixed-element meshes in three dimensions. The main contributions of this paper are three-fold. First, we present a framework for creating geometrically exact meshes comprised of rational Bézier hexahedra, tetrahedra, wedges and pyramids through degree elevation of suitable linear meshes. We additionally discuss how our approach may be applied to higher-order mesh generation for more traditional finite element approaches. Next, we propose two quality metrics for three-dimensional curvilinear meshes, and discuss some important considerations for mesh quality of higher-order meshes. Finally, we demonstrate the analysis suitability of the meshes constructed using our framework through numerical examples. [ABSTRACT FROM AUTHOR]

This paper designs and analyzes a new and stable Petrov–Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov–Galerkin immersed finite element method uses the immersed finite element functions for the trial space and the standard finite element functions for the test space. Both the a priori and a posteriori error estimates of the method are analyzed in this paper. We prove the continuity and inf-sup condition and the a priori error estimate of the energy norm. The proposed a posteriori error estimator is proved to be both reliable and efficient, with both reliability and efficiency constants independent of the location of the interface. Extensive numerical results confirm the numerical scheme's optimal convergence and indicate the robustness with respect to the interface-mesh intersection and the coefficient contrast, despite the robustness of the inf-sup constant with respect to the interface-mesh intersection has yet been theoretically proved. [ABSTRACT FROM AUTHOR]

The use of Electro-Active Polymers (EAPs) for the fabrication of evermore sophisticated miniaturised soft robotic actuators has seen an impressive development in recent years. The incorporation of crystallographic anisotropic micro-architectures, within an otherwise nearly uniform isotropic soft polymer matrix, has shown great potential in terms of advanced three-dimensional actuation (i.e. stretching, bending, twisting), especially at large strains, that is, beyond the onset of geometrical pull-in instabilities. From the computational point of view, the design of accurate and robust albeit efficient constitutive models is a very active area of research. This paper introduces a novel polyconvex phenomenological invariant-based transversely isotropic formulation (and relevant computational frameworks) for the simulation of transversely isotropic EAPs at large strains, where the ab initio satisfaction of polyconvexity is exploited to ensure the robustness of numerical results for any range of deformations and applied electric fields. The paper also presents key important results both in terms of the existence of minimisers and material stability of coupled electro-mechanics, enhancing previous works in the area of large strain elasticity. In addition, a comprehensive series of selected numerical examples is included in order to demonstrate the effect that the anisotropic orientation and the contrast of material properties, as well as the level of deformation and electric field, have upon the response of the EAP when subjected to large three-dimensional stretching, bending and torsion, including the possible development of wrinkling and the potential loss of ellipticity in ill-posed constitutive models. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) have recently been used to solve various computational problems which are governed by partial differential equations (PDEs). In this paper, we propose a multi-output physics-informed neural network (MO-PINN) which can provide solutions with uncertainty distributions for both forward and inverse PDE problems with noisy data. In this framework, the uncertainty arising from the noisy data is first translated into multiple measurements regarding the prior noise distribution using the bootstrap method, and then the outputs of neural networks are designed to satisfy the measurements as well as the underlying physical laws. The posterior estimation of target parameters can be obtained at the end of training, which can be further used for uncertainty quantification and decision making. In this paper, MO-PINNs are demonstrated with a series of numerical experiments including both linear and nonlinear, forward and inverse problems. The results show that MO-PINN is able to provide accurate predictions with noisy data. In addition, we also demonstrate that the prediction and posterior distributions from MO-PINNs are consistent with the solutions from traditional a finite element method (FEM) solver and Monte Carlo methods given the same data and prior knowledge. Finally, we show that additional statistical knowledge can be incorporated into the training to improve the prediction if available. [ABSTRACT FROM AUTHOR]

This paper presents a model-free data-driven strategy for linear and non-linear finite element computations of open-cell foam. Employing sets of material data, the data-driven problem is formulated as the minimization of a distance function subjected to the physical constraints from the kinematics and the balance laws of the mechanical problem. The material data sets of the foam are deduced here from representative microscopic material volumes. These volume elements capture the stochastic characteristics of the open-cell microstructure and the properties of the polyurethane material. Their computation provides the required stress–strain data used in the macroscopic finite element computations without postulating a specific constitutive model. The paper shows how to derive suitable material data sets for the different cases of (non-)linear and (an-)isotropic material behavior efficiently. Exemplarily, we compare data-driven finite element computations with linearized and finite deformations and show that in a data-driven computation the linear kinematic is sufficiently accurate to capture the material's non-linearity up to 50% of straining. The numerical example of a rubber sealing illustrates possible areas of application, the expenditure, and the proposed strategy's versatility. [ABSTRACT FROM AUTHOR]