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This paper presents a stability-preserving model reduction approach for a vibro-acoustic finite element model including poroelastic materials. Most of the research on these systems in the past was conducted in the frequency domain and there were less focus on the stability properties. However, with the increasing of interest in time-domain auralization and virtual sensing, stability-preserving model order reduction is becoming essential for efficient time-domain simulations. The original finite element models for such systems are already well established but not extended to the time domain because of its high computational demand. Therefore, this paper proposes a method to generate stable reduced-order models. The solid displacement–total displacement ( u s − u t ) formulation of the Biot's model is used to describe the poroelastic media. This formulation leads to a set of positive and symmetric system matrices with passive transfer functions. Applying the Kalman–Yakubovich–Popov lemma and the Cholesky/LDL decomposition, this formulation is modified to satisfy the previously proven stability-preserving conditions under one-sided model order reduction. Furthermore, the coupling conditions between the poroelastic media, air, and structure are also investigated. It is finally shown that the stability-preserving property is kept for the coupled models with appropriate variables. The proposed method is verified by several numerical simulations. • A time-domain formulation of Biot's equations is derived. • The time-domain Biot's model is stability-preserving under one-sided model order reduction methods. • The coupling models between the proposed time-domain Biot's model and vibro-acoustic systems are still stability-preserving. • The reduced order model enables efficient time-domain simulations of the vibro-acoustic system with the Biot's model. [ABSTRACT FROM AUTHOR]

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]

Shape memory alloys are remarkable 'smart' materials used in a broad spectrum of applications, ranging from aerospace to robotics, thanks to their unique thermomechanical coupling capabilities. Given the complex properties of shape memory alloys, which are largely influenced by thermal and mechanical loads, as well as their loading history, predicting their behavior can be challenging. Consequently, there exists a pronounced demand for an efficient material model to simulate the behavior of these alloys. This paper introduces a material model rooted in Hamilton's principle. The key advantages of the presented material model encompass a more accurate depiction of the internal variable evolution and heightened robustness. As such, the proposed material model signifies an advancement in the realistic and efficient simulation of shape memory alloys. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical modeling of gas flow in porous media with compressible gas and rock. To develop an effective numerical method for simulating this problem, we propose an alternative equation by introducing the poroelasticity equation and the porosity variation equation to account for the influence of rock deformation on porosity. In addition, we introduce a free energy for the skeleton of rocks and take into account rock compressibility as well. Instead of the pressure gradient, we utilize the chemical potential gradient as the primary driving force. This formulation is proved to satisfy an energy dissipation law. By applying the improved energy factorization method to handle the Helmholtz free energy density, we propose a semi-implicit time discretization scheme. The discrete pressure is carefully calculated by the discrete chemical potential and Helmholtz free energy so as to inherit the energy dissipation law. A fully discrete scheme is constructed based on the discontinuous Galerkin and mixed finite element methods with the upwind strategy. We prove that the fully discrete scheme still satisfies the energy dissipation law as well as possesses the feature of mass conservation. Additionally, we can prove the boundedness of density under reasonable assumptions on porosity. Numerical results are provided to show the performance of the proposed scheme and validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper introduces a novel computational framework, Finite Element Method Integrated Networks (FEMIN), designed to accelerate crash simulations significantly. The core innovation is that large regions of the mesh in a Finite Element Method (FEM) crash simulation are replaced by a Neural Network (NN). The NN is directly integrated into the FEM solver and aims to reduce computational time and resources while maintaining high-fidelity results. The introduced NN architectures and training strategies address the inherent challenges posed by crash simulations, such as long time series, dynamic loads, and complex, history-dependent material behaviors. Our work includes the development of load cases tailored to validate the FEMIN approach. Furthermore, we introduce a novel NN architecture, the Temporal gated Multi-Layer Perceptron (TgMLP), both as a stand-alone model and also as an initial state predictor in combination with a conditional Long Short-Term Memory (LSTM) network and compare their effectiveness in FEMIN. The results demonstrate that FEMIN, employing these NN models, can accurately predict the structural response under crash conditions while inferring the loads significantly faster than the replaced FEM mesh. This research paves the way for more efficient vehicle design, offering a promising solution for integrating machine learning into computational mechanics. [ABSTRACT FROM AUTHOR]

The Dual Mesh Control Domain Method (DMCDM), proposed and developed by the second author, is a recent advancement in the field of numerical methods. This method represents a blend of the Finite Element Method (FEM) and the Finite Volume Method (FVM), each having its primary domains of popularity. The DMCDM incorporates the concepts of interpolation, duality, and numerical integration from the former and the idea of satisfying the integral (global) form of the differential (or balance) equations from the latter. In this paper, we present a fresh and systematic approach to solving two-dimensional nonlinear problems, including advection–diffusion problems, with arbitrary primal meshes. We develop an approach where element-level equations are derived and assembled in a manner similar to that of the FEM, as opposed to using nodal equations, as has been the convention in all of the previous works involving the DMCDM. Additionally, we extend the previous works to include higher-order interpolation, specifically quadratic interpolation. In short, we provide formulations for four-node bilinear and nine-node biquadratic elements of quadrilateral shape, enabling the application of the DMCDM to arbitrary primal and dual meshes. Finally, we explore the Newton iteration method for linearizing the nonlinear system of equations that arises from the DMCDM discretization. We present numerical results and comparisons against available exact and numerical solutions to verify the accuracy and robustness of the DMCDM. [ABSTRACT FROM AUTHOR]

This paper presents the basis of a new mechanical model named QuadWire dedicated to efficient simulations of bead-based additive manufacturing processes in which elongated beads undergoing significant cooling and eigenstrain are assembled to form 3D parts. The key contribution is to use a multi-particular approach containing 4 particles per material point to develop an extended 1D model capable of capturing complex 3D mechanical states, while significantly reducing computation time with respect to conventional approaches. Indeed, 3D models usually require at least 3 to 4 elements across the bead section, which results in fine discretization along the tangential direction to avoid conditioning issues, and therefore very fine mesh of the entire 3D part. In the QuadWire model, the bead height and thickness are internal dimensions, enabling a significantly coarser mesh along the tangential direction. Thus, although the QuadWire has 12 degrees of freedom per material point instead of 3 for classical models, the total number of degrees of freedom is reduced by several orders of magnitude for large parts. The proposed model is classically developed within the framework of the principle of virtual power and standard generalized hyperelastic media (i.e., finite strain theory), which necessitates a thermodynamic analysis. Furthermore, the proposed approach includes native and manageable kinematic constraints between successive beads so that the stress state properly evolves during fabrication. Finite element analysis is used for numerical implementation under infinitesimal strain assumption for the sake of simplicity, and the QuadWire stiffness parameters are optimized so that the mechanical response fit conventional 3D approaches. To validate and demonstrate the capabilities of the proposed strategy, the evolution of displacements and stresses in fused deposition modeling of polylactide is simulated. [ABSTRACT FROM AUTHOR]

In this paper, an enhanced prismatic-element scaled boundary perfectly matched layer (SBPML) is developed, which is a novel time-domain artificial boundary method for 3D infinite wave problems. The SBPML permits the utilization of an artificial boundary with general geometry and can consider planar physical surfaces and interfaces extending to infinity. Moreover, this enhancement enables the seamless integration with surface meshes of arbitrary polygonal faces (such as triangles, quadrangles, pentagons, heptagons, octagons, etc.) at the truncation boundaries. The interior domain can be modelled by polyhedrons formulated by the scaled boundary finite element method (SBFEM). The geometric shape of the SBPML elements in the proposed method, within the local coordinate system, takes the form of a regular prism with regular polygonal top and bottom faces. Consequently, it is referred to as the Prismatic-element SBPML. Local scaled boundary coordinates based on the polygonal shape function are firstly introduced into the truncated infinite domain to describe its general geometry properties and provide the coupled capabilities with arbitrary polygonal face meshes at the truncation boundaries of numerical models. Then, a complex stretching function from PML is applied to radial direction of the local scaled boundary coordinates to map the physical space onto the complex space, resulting in a SBPML domain. Upon spatial discretization, the proposed method is formulated using a 2nd-order mixed displacement-stress unsplit-field formulation, allowing seamless coupling with FEM and SBFEM originally designed for the interior computational domain. Finally, three benchmark examples of wave propagation problems in unbounded domains and two soil-structure interaction problems of engineering problems are presented to demonstrate the accuracy and robustness of the Prismatic-element SBPML. [ABSTRACT FROM AUTHOR]

In this paper, a computational framework based on the Smoothed Particle Finite Element Method is developed to study the coupled seepage-deformation process in unsaturated porous media. Governing equations are derived from the balance laws of solid and fluid phases considering partial saturation effects in porous media. Moreover, an hourglass control method is implemented to avoid the rank-deficiency issue in SPFEM and the moving least squares approximation technique (MLS) is implemented to eliminate the pore pressure oscillations when the low-order triangle element is used. The proposed coupled SPFEM formulation is validated through four elastic examples and one elasto-plastic example. Good agreement with the numerical or analytical results reported in the literature is obtained. Further, the rainfall-induced slope failure is studied, in which a suction-dependent elasto-plastic Mohr–Coulomb model is adopted to take account of the suction effect in unsaturated soil. The evolution of the suction and soil deformation during the rainfall period and the whole slope failure process are obtained. It is demonstrated that the proposed method is a promising tool in numerical investigations of both the triggering mechanisms and post-failure behavior of the rainfall-induced slope failure. [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]

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 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 develop a partition-of-unity construction of the stabilization function required in Nitsche's method, which can be seen as a generalization of the element-wise construction that is widely used in finite element methods. This allows for the use of Nitsche's method within the Partition of Unity Method with a stabilization function that is not simply a constant over the whole boundary. In addition to that, we introduce a patch-aggregation approach designed to avoid arbitrarily large values of the stabilization function and the associated ill-conditioned systems and deteriorated convergence rates. We present numerical results to validate the proposed methods, covering Dirichlet boundary conditions, interface constraints and higher-order problems. These results clearly show that our approach leads to optimal convergence rates. [ABSTRACT FROM AUTHOR]

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier–Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov–Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes. • The use of machine learning libraries to solve discretised systems of PDEs exactly. • The first application of this idea to multiphase flows with interface-capturing methods. • The code can run, with no modification, on CPUs and GPUs. • It also runs, with no modification, on the latest energy-efficient AI processors. • A new compressive advection scheme sharpens the interface between the fluids. [ABSTRACT FROM AUTHOR]

In this paper, a new framework for large strain of electro-active viscoelastic polymeric beams is developed. The kinematical quantities of beam are derived, and then the constitutive equations of electromechanical beam are developed. To expand the formulation to viscoelastic regime, a generalization of quasi-linear viscoelasticity theory for electo-mechanical deformation is developed and called electro-mechanical quasi-linear viscoelastic (EM-QLV) theory. To deal with highly nonlinear formulation, a finite element formulation in Total Lagrangian framework is derived. Finally, to examine the applicability of the proposed model, several examples are presented and compared with the 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]

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]

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]

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]

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 presents a structural optimisation method in three-dimensional acoustic–elastic coupled problems. The proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points. In the process of the optimisation, configuration of the elastic materials is expressed with a level set function, and the distribution of the level set function is iteratively updated with the help of the topological derivative. The topological derivative is associated with state and adjoint variables which are the solutions of the acoustic–elastic coupled problems. In this paper, the acoustic–elastic coupled problems are solved by a BEM–FEM coupled solver, in which the fast multipole method (FMM) and a multi-frontal solver for sparse matrices are efficiently combined. Along with the detailed formulations for the topological derivative and the BEM–FEM coupled solver, we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves, from which we confirm the effectiveness of the present method. [ABSTRACT FROM AUTHOR]

This paper introduces a new method for the integration of localized surface geometry with fully parameterized absolute nodal coordinate formulation (ANCF) finite elements. In this investigation, ANCF finite elements are used to create the global geometry and perform the finite element (FE)/multibody system (MBS) analysis of deformable bodies. The localized surface geometry details can be described on ANCF element surfaces without the need for mesh refinement. The localized surface is represented using a standard computational geometry method, Non-uniform rational B-spline surface (NURBS), which can describe both conic surface and freeform surface efficiently and accurately. The basic idea of the integration of localized surface geometry with ANCF elements lies in the inclusion of such detail in the element mass matrix and forces. The integration can be implemented by overlapping the localized surface geometry on the original ANCF element or by directly trimming the ANCF element domain to fit the required shape. During the integration process, a mapping between ANCF local coordinates and NURBS localized geometric parameters is used for a consistent implementation of the overlapping and trimming methods. Additionally, two numerical integration methods are compared for the rate of convergence. The results show that the proposed subdomain integration method is better, since it is optimized for dealing with complex geometry. The proposed subdomain method can be used with any fully parameterized ANCF element. In order to analyze the accuracy of the proposed method, a cantilever plate example with localized surface geometry is used, and the simulation results with the proposed method are compared with the simulation results obtained using a commercial FE code. Two other examples that include contact with ground and localized surface geometry are also provided. These examples are a simple plate structure with surface geometry and a tire with tread details. The incompressible hyperelastic Mooney–Rivlin material model is used to describe the material used in the tire tread. It is shown through the tire contact patch that the proposed method can successfully capture the effect of the tread grooves. The rate of convergence and locking of fully parameterized ANCF elements are also discussed in this paper. [ABSTRACT FROM AUTHOR]

We in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures. The key idea is to develop a general refinement algorithm based on a so-called polytree mesh structure. The method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones, which are considered as special cases. Also, polytree meshes are conforming and can be regarded as a generalization of quadtree meshes. For the aim of this paper, we restrict our main interest in plane-strain limit analysis to von Mises-type materials, yet its extension to a wide class of other solid mechanics problems and materials is completely possible. To avoid volumetric locking, we propose an approximate velocity field enriched with bubble functions using Wachspress coordinates on a primal-mesh and design carefully strain rates on a dual-mesh level. An adaptive mesh refinement process is guided by an L 2 -norm-based indicator of strain rates. Through numerical validations, we show that the present method reaches high accuracy with low computational cost. This allows us to perform large-scale limit analysis problems favorably. [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]

In this paper, we study the performance of the Generalized Finite Element Method (GFEM) applied to the Poisson problem with crack singularities. Recently, a GFEM with modified Heaviside enrichments was proposed in Gupta et al. (2013) to approximate the solution of a 2D elasticity problem with a crack. It was shown that the GFEM is indeed a Stable GFEM (SGFEM), i.e., it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method (FEM). However, the robustness of the conditioning of the GFEM with respect to the position of the mesh relative to the crack was not addressed in Gupta et al. (2013). In this paper, we observed that the conditioning of the GFEM with the enrichments used in Gupta et al. (2013) is not robust when applied to approximate the solution of the Poisson problem. Moreover, the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces. We proposed using additional singular enrichments at the nodes close to the crack near the crack-tip. We proved that the GFEM, with enrichments proposed in this paper, yields optimal order of convergence irrespective of the position of the mesh. Moreover, with a local orthogonalization procedure, we have shown through numerical experiments that the conditioning of this GFEM is not worse than that of the standard FEM and the conditioning is robust with respect to the position of the mesh. Thus the GFEM, with the enrichments suggested in this paper, is indeed an SGFEM when applied to a Poisson problem with the crack singularities. [ABSTRACT FROM AUTHOR]

In this paper, we present an approximation of elliptic problems with multiscale and high-contrast diffusion coefficients. A mixed formulation is considered such that both pressure and velocity are approximated simultaneously. This formulation arises naturally in many applications such as flows in porous media. Due to the multiscale nature of the solutions, using model reduction is required to efficiently obtain approximate solutions. There are many multiscale approaches for elliptic problems in mixed formulation. These approaches include numerical homogenization and mixed multiscale finite element method, which aim to obtain a coarse accurate representation of the velocity without using an accurate representation for pressure. It has been a challenging task to construct a method giving accurate representation for both pressure and velocity. The goal in this paper is to construct multiscale basis functions for both pressure and velocity. We will apply the framework of Generalized Multiscale Finite Element Method (GMsFEM), and design systematic strategies for the construction of basis. The construction involves snapshot spaces and dimension reduction via local spectral problems. The mixed formulation is minimized in the sense of least-squares. The compatibility condition for multiscale finite element spaces of the pressure and velocity is not required in the least-squares mixed form. This gives more flexibility for the construction of multiscale basis functions for velocity and pressure. Convergence analysis is carried out for the least-squares mixed GMsFEM. Several numerical examples for various permeability fields are presented to show the performance of the presented method. The numerical results show that the least-squares mixed GMsFEM can give accurate approximation for both pressure and velocity using only a few basis functions per coarse element. [ABSTRACT FROM AUTHOR]

With Adaptively Weighted (AW) numerical integration, for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain, and then are corrected for the deviation from the original domain using shape sensitivity analysis. In this paper, we demonstrate the application of AW integration scheme in the context of the Finite Cell Method which must perform numerical integration over arbitrary domains without meshing. The standard integration technique employed in FCM is the characteristic function method that converts the continuous integrand over a complex domain into a discontinuous integrand over a simple (box) domain. Then, well known integrand adaptivity techniques are employed to integrate the resulting discontinuous integrand over the box domain. Although this method is simple to implement, it becomes computationally very expensive for realistic complex 3D domains such as sculptures, bones and engines. In contrast, in AW scheme the quadrature weights directly adapt to the complex geometric domain without the need to making the integrand discontinuous leading to superior computational properties. In this paper, we demonstrate the computational efficiency of AW over the characteristic function method as it requires fewer subdivisions and less time to achieve a given accuracy in both two and three dimensions. In addition, AW offers a number of advantages including flexibility in the choice of quadrature points and basis functions. [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]