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Transverse shear locking is an issue that occurs in Reissner–Mindlin plate and shell elements. It leads to an artificial stiffening of the system and to oscillations in the stress resultants for thin structures. The thinner the structure is, the more pronounced are the effects. Since transverse shear locking is caused by a mismatch in the approximation spaces of the displacements and the rotations, a field-consistent approach is proposed for an isogeometric degenerated Reissner–Mindlin shell formulation. The efficiency and accuracy of the method is investigated for benchmark plate and shell problems. A comparison to element formulations with locking alleviation methods from the literature is provided.

This paper presents a variable cutting (VCUT) level set based topology optimization method to design functionally graded cellular structures (FGCS). A variable and continuous cutting function by interpolating with a set of height variables is proposed to generate functionally graded cellular structures, which offers a novel tool to optimize the macroscopic graded pattern. Due to the continuity of the cutting function, perfect geometric connections between adjacent cells are guaranteed without imposing extra constraints in the optimization. In addition, a solid covering skin can be easily attached to the FGCS by means of the variable cutting function. Three FGCS design problems, including graded density control, compliance minimization and layered cellular structure design, are presented to demonstrate the effectiveness and applicability of the proposed method. The optimized 2D and 3D macroscopic FGCS designs exhibit well-performing structural layout with fully connected micro-scale geometries.

Additively manufactured lattice structures enable the realisation of light-weight, multi-functional, structures. For example, lattices can be used for high stiffness and buckling resistance in sandwich structures or as support material for additive manufacturing. Topology optimisation and additive manufacturing are two technologies that allow the design, optimisation and manufacture of complex lattice designs. In this work, a new lattice optimisation methodology is presented that tailors the size, shape and orientation of individual lattice trusses in three-dimensional space by using principal strain fields obtained from topology optimisation. This new method of generating functionally graded lattices is shown both numerically and experimentally to be capable of generating lattice structures with greatly improved stiffness and strength when compared to lattice structures with a uniform lattice infill. Upper and lower relative density thresholds and minimum truss member sizes are included in the optimisation workflow to ensure that the optimised lattice designs are compatible with additive manufacturing process constraints. The functional grading method is also shown to be capable of generating conformal lattice structures in three dimensions, even for complex loading conditions and arbitrary volume boundaries.

A three-dimensional (3D) isogeometric/discrete-element coupling method is presented for modeling contact/impact between structures and particles. This method takes advantages of the geometry smoothness and exactness of isogeometric analysis (IGA) for continuous solid media and the effectiveness and flexibility of the discrete element method (DEM) for particulate matters. The coupling procedure for handling interactions between IGA elements and discrete elements (DEs) includes global search, local search and interaction calculation. In the global search, the CGRID method is modified to detect potential contact pairs between IGA elements and DEs based on their bounding box representations. The strong convex hull property of a NURBS control mesh plays an important part in the bounding box representation of IGA elements. In the local search, the proposed approach treats each spherical DE centroid as a slave node and the contact surface of each IGA element as the master surface. The projection of a DE centroid onto an IGA element contact surface is solved by modifying the simplex method and Brent iterations. The contact force between an IGA element and a DE is determined from their penetration by using a (nonlinear) penalty function based method. The whole coupled system is solved by the explicit time integration within a updated Lagrangian scheme. Finally, three impact examples, including the impact of two symmetric bars, a tube onto a footing strip, and an assembly of granular particles to a tailor rolled blank, are simulated in elastic regime to assess the accuracy and applicability of the proposed method.

We investigate, both mathematically and numerically, the self-consistent clustering analysis recently introduced by Liu–Bessa–Liu and, independently, by Wulfinghoff–Cavaliere–Reese. We establish, in the small strain setting and non-softening material behavior, existence and uniqueness of the solution to the discretized equations for fixed (possibly anisotropic) reference material, by a constructive method. Furthermore, we establish convergence of the solution to the discretized equation to the continuous solution upon refinement of the clusters. Thus we generalize the work of Tang–Zhang–Liu to spatial dimensions larger than 1. We explore the specific structure of the Lippmann–Schwinger equation governing clustering analysis, proving strict equivalence to a problem of Eyre–Milton type. For the latter formulation, existence and uniqueness are easily established based on recent progress in the understanding of polarization schemes in FFT-based computational homogenization methods. Last but not least, for elasto-viscoplastic constituents we clarify the relationship of the self-consistent clustering analysis to the transformation field analysis of Dvorak–Benveniste. Our theoretical considerations are confirmed by pertinent numerical experiments.

QM (quantum mechenics) and MM (molecular mechenics) coupling methods are widely used in simulations of crystalline defects. In this paper, we construct a residual based a posteriori error indicator for QM/MM coupling approximations. We prove the reliability of the error indicator (upper bound of the true approximation error) and develop some sampling techniques for its efficient calculation. Based on the error indicator and D\"{o}rfler marking strategy, we design an adaptive QM/MM algorithm for crystalline defects and demonstrate the efficiency with some numerical experiments., Comment: 21 pages, 9 figures

In order to model the fatigue crack propagation of engineering structures with high efficiency, a novel multi-grid (MG) reanalysis method is formulated in this study under the framework of the extended finite element method (XFEM). In the numerical approach, the MG is firstly used to establish connections between the mesh with no crack and the meshes with crack. The transfer operators are constructed only in the initial MG analysis and will be reused in the crack propagation analysis. Then, the equilibrium equations of crack propagation can be efficiently solved through the MG iterations. Several numerical examples are investigated to verify the validity of this method and it is shown that the presented algorithm is very accurate and can efficiently reduce the computational cost.

In the past the Vertex Morphing method has mainly been used in the context of optimization of thin walled shell structures or fluid mechanical problems. In this paper an extension of this parameter free shape optimization method to 3-dimensional solid structures based on the free Finite Element code CalculiX is presented. Key element of the whole process is an efficient adjoint sensitivity analysis that can handle many design variables providing gradient information for the response functions of interest, e.g. stresses and mass. Typically, parameter free shape optimal designs result in non-smooth geometries and suffer from mesh dependencies. To avoid these phenomena an intermediate filter operation in combination with several post processing steps of the raw sensitivities is added. In the optimization, the filter size is also used as an additional design parameter controlling the curvature of the optimal shape. In many engineering optimization applications constraints have to be considered. These restrictions are included in the optimization problem by the use of Rosen’s gradient projection method. For the successive shape update a pseudo linear-elastic static Finite Element analysis with multiple point constraints and a varying Young’s modulus is performed in every design iteration . Finally, the capabilities of the extended Vertex Morphing method are demonstrated for an academic example and a geometrically complex part from aerospace industry.

In terms of the three-field (u-w-p ) formulation of Biot’s theory of saturated porous media with incompressible solid and fluid phases, the numerical manifold method (NMM) models are developed to analyze the fully dynamic consolidation of fractured porous media in this study. The same approximation to fluid velocity and skeleton displacement is constructed which is capable of modeling both incompressible and compressible deformation, while two types of approximations to pore pressure field are established. Since the inertial effect of fluid is not neglected, the proposed model can fully capture the dynamic behavior of porous media, especially under the impact or high-frequency loading condition and, accordingly, exhibits apparent superiority in predicting transient and wave propagation responses of cracked porous media. Moreover, low order interpolation functions for primal variables and the most flexible three-node triangular finite element mesh are used, which both are difficult to implement using other partition of unity (PU) based numerical methods in terms of Biot’s two-field formulation. Also, the discontinuities can be modeled more naturally in the NMM framework in comparison with XFEM or PNM . Meanwhile, an augmented Lagrange multiplier method for stick–slip contact model is first incorporated into fully-dynamic three-field Biot’ formulation. In addition, a mass lumping technique within NMM framework, which turns out to be a unique advantage of NMM over other numerical methods, is employed to suppress unphysical oscillations and increase computational efficiency. Energy balance condition is employed to evaluate the stability and accuracy of the time integration scheme. The robustness and versatility of the proposed models are manifested with several typical examples.

This paper presents a topology optimization approach to design support structures with efficient thermal conductive capabilities for powder-bed based laser additive manufacturing (AM). A transient heat transfer model is proposed to simulate the temperature distribution of the point-by-point and layer-upon-layer AM process. The process model is seamlessly integrated into a density based structural topology optimization method with an adjoint based sensitivity analysis and a gradient based optimization. For a given prototype shape, the layout of the corresponding support structure is designed such that the thermal energy during the fabrication process can be efficiently transferred from the laser path to a prescribed heat sink. Besides, an overhang constraint is proposed and used together with an AM filter to avoid overhang features in both the optimized support structure and the prototype, ensuring the overall manufacturability. Numerical examples and discussions are given to demonstrate its applicability.

We combine decision theory with fundamental stochastic time-optimal path planning to develop partial-differential-equations-based schemes for risk-optimal path planning in uncertain, strong and dynamic flows. The path planning proceeds in three steps: (i) predict the probability distribution of environmental flows, (ii) compute the distribution of exact time-optimal paths for the above flow distribution by solving stochastic dynamically orthogonal level set equations , and (iii) compute the risk of being suboptimal given the uncertain time-optimal path predictions and determine the plan that minimizes the risk. We showcase our theory and schemes by planning risk-optimal paths of unmanned and/or autonomous vehicles in illustrative idealized canonical flow scenarios commonly encountered in the coastal oceans and urban environments. The step-by-step procedure for computing the risk-optimal paths is presented and the key properties of the risk-optimal paths are analyzed.

We investigate a novel technique to obtain the mesh porosity (volume and area) based on the Gilbert–Johnson–Keerthi (GJK) distance algorithm for flow modelling. The GJK algorithm is used to check collisions between two convex objects. This concept is applied to check the collision between an element of the computational mesh and the geometrical model where the element of the mesh is broken recursively to obtain refined values of the porosity. The approach has demonstrated that it can handle complex geometries within feasible computational time. The porosity values obtained using this methodology are coupled in the developed Navier–Stokes. Numerical findings are compared with selected benchmarking tests. Numerical findings agree with experimental data and the main features of the fluid flow are well captured.

We present a new multidimensional high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher-order elements. We also present new one-dimensional test problems that have an analytic solution corresponding to a hyperelastic–plastic wave. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated high-order Taylor series polynomials. The stress fields are represented using nodal quantities. The constitutive models used to calculate the deviatoric stress are either a hypoelastic–plastic, infinitesimal strain hyperelastic–plastic, or finite strain hyperelastic–plastic model. These constitutive models require new methods for calculating high-order polynomials for the velocity gradient and deformation gradient in an element. The plasticity associated with the strength model is determined using a radial return method with a J 2 yield criterion and perfect plasticity. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. A diverse suite of 1D and 2D test problems are calculated. The new 1D piston test problems, which have analytic solutions for each elastic–plastic model, are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new Lagrangian DG method. 2D test problems are calculated to demonstrate the stability and robustness of the new Lagrangian DG method on multidimensional problems with high-order elements, which have faces that can bend.

The advent of fabrication techniques such as additive manufacturing has focused attention on the considerable variability of material response due to defects and other microstructural aspects. This variability motivates the development of an enhanced design methodology that incorporates inherent material variability to provide robust predictions of performance. In this work, we develop plasticity models capable of representing the distribution of mechanical responses observed in experiments using traditional plasticity models of the mean response and recently developed uncertainty quantification (UQ) techniques. We demonstrate that the new method provides predictive realizations that are superior to more traditional ones, and how these UQ techniques can be used in model selection and assessing the quality of calibrated physical parameters., 21 pages, 6 composite figures. arXiv admin note: substantial text overlap with arXiv:1802.01487

The ability to make decisions based on quantities of interest that depend on variables inferred from measurement finds application in different fields of mechanics and physics. The evaluation of the inferred variables, and hence the quantities of interest, from the measurement typically requires the solution of an inverse problem . For example, in medical imaging the elastic heterogeneity of a tumor and its nonlinear elastic response can be used to distinguish benign tumors from their malignant counterparts. These images of linear and nonlinear elastic parameters of tissue are typically obtained by using a measured displacement field and solving a complex inverse elasticity problem . In this paper we consider circumventing the solution of the inverse problem by using measured displacements as input to a deep convolutional neural network (CNN) and training it to classify tumors on the basis of their elastic heterogeneity and nonlinearity. For a simple, 5-layer CNN trained with 8,000 samples for heterogeneity, and a 4-layer CNN trained with 4,000 samples for nonlinear elasticity we report classification accuracies in the range of 99 . 7 % − 99 . 9 % . The training and testing data are both obtained from the forward solution of finite element models of samples. We also analyze the weights of the trained model to understand the process through which the network extracts features of elastic moduli from the input displacement images. Finally, we apply the nonlinear elasticity classifier , which is trained entirely using simulated data, to displacement images obtained from ten patients with breast lesions and note that it correctly classifies eight out of ten cases. This application illustrates how data from physics-based models can be used in improving the performance of a data-driven algorithm in data-sparse scenarios.

There is growing interest in high fidelity simulations of buoyancy-driven flows in indoor environments. This is driven to a large extent by the push to utilize naturally occurring flows to reduce energy consumption in the built environment. Naturally occurring (buoyancy-driven) flows exhibit spatiotemporal thermal fluctuations, and accurately capturing these thermal variations is essential to calculate energy usage. In this work, we deploy a residual-based variational multiscale (VMS) finite element large-eddy simulation model for accurately modeling buoyancy-driven flows in enclosed environments. We use the canonical example of the Rayleigh–Benard convection problem in 2D and 3D to verify and validate the VMS model. We show good comparison with benchmark numerical and experimental results across seven orders of magnitude variation in Rayleigh numbers ( R a ∼ 1 0 3 to 1 0 10 ), covering laminar, transition, and turbulent regimes without any extra treatments. We additionally employ weak enforcement of Dirichlet boundary conditions for both velocity and temperature, and show that comparable results can be produced with much coarser meshes. This confirms that the VMS framework with weak imposition of Dirichlet boundary conditions is a computationally efficient approach to model buoyancy-driven flow physics in complex indoor environments.

Non-invasive estimation of fractional flow reserve (FFR) values, the key index in the diagnosis of obstructive coronary artery disease, is a promising alternative to traditional way of performing invasive coronary angiography. With the advances in computational fluid dynamics (CFD), one can estimate FFR based on the solution obtained in a reconstructed coronary geometry from coronary computed tomography (CT) angiography. However, the computational cost to perform three-dimensional (3D) simulations has limited the use of CFD in most clinical settings. This could become more restrictive if one aims to quantify the uncertainty associated with FFR calculations due to the uncertainty in anatomic and physiologic properties as a significant number of 3D simulations is required to sample a relatively large parametric space . We have developed a predictive probabilistic model of FFR, which quantifies the uncertainty of the predicted values with significantly lower computational costs. Based on global sensitivity analysis, we first identify the important physiologic and anatomic parameters that impact the predictions of FFR. Our approach is to employ one-dimensional blood flow simulations of coronary trees that offer fast FFR predictions with uncertainty quantification in computing blood pressure and flow distributions within the coronaries. This is complemented with a multifidelity algorithm that is used to infer their optimal values using available patient-specific clinical measurements.

One of the main challenges for multiphysics simulations of volume-coupled problems is data exchange, which becomes a serious bottleneck for large-scale applications executed on distributed memory computer architectures. Unlike surface coupling, the transfer of volumetric information via network communication can be too inefficient. To circumvent this problem, a new concept based on a joint hierarchical mesh is proposed, where the coupled solvers use different cells of a single shared grid. A domain decomposition method based on space-filling curves guarantees that all coupling data can be exchanged locally by in-memory operations. Furthermore, the interleaved execution of the solvers ensures that there is no additional overhead due to communication barriers. To demonstrate the efficiency of the presented method for realistic three-dimensional problems, it is used to predict the acoustic field of a turbulent jet in a direct-hybrid simulation, in which fluid dynamics and aeroacoustics are directly coupled. The results confirm the good parallel scalability of the approach and show the design of the coupling algorithm to be more efficient than a standard hybrid flow-aeroacoustics scheme coupled via disk I/O.

The majority of the current structural optimization software does not accommodate manufacturing constraints. Therefore, substantial modifications are imposed upon optimized designs to make them manufacturable and hence nonoptimal. We propose to optimize the design of composite structures that are amenable to Additive Manufacturing (AM). The printing process chosen in our study is based on Direct Ink Writing (DIW) in which short carbon fibers in a thermoset resin are extruded through a moving nozzle to build up a structure. Since the fibers are primarily aligned in the flow direction of the extrudate, the DIW printing trajectory influences the material properties of the composite structure. To accommodate this, the extrudate trajectory follows the contours of parameterized level-set functions. The parametrization allows us to prescribe the material properties and impose many DIW manufacturing constraints such as no-overlap, no-sag, minimum radius of curvature and continuity of the toolpaths. Ultimately, we obtain optimal manufacturable toolpaths that start and finish at a boundary. To minimize the fabrication time, we formulate the linking sequence of the toolpaths as a traveling salesman problem which we solve to obtain the shortest continuous toolpath per layer. Several examples illustrate the optimization procedure. Validation is also performed.

In this paper, a new fully-resolved framework capable of capturing the fundamental physics of particle–fluid interactions, the collision of particles with solid surfaces , and the resulting damage is proposed. A coupled DEM-IB-CLBM, consisting of a discrete element method (DEM), an immersed boundary (IB) method, and a cascaded lattice Boltzmann method (CLBM), is used to fully resolve the interaction of the particles with the surrounding viscous fluid . The peridynamics theory is then implemented and used to predict the impact damage to the target material. This framework is validated by comparing the trajectory of a particle–wall collision event in a viscous fluid with the previous results in the literature. Furthermore, the variation of the restitution coefficient with the impact velocity is in a good agreement with the available experimental results. The influence of multiple impacts and the resulting surface damage on the fluid dynamics of the system is investigated. It is demonstrated that the method correctly predicts the expected effects of multiple collisions and impact angle variations on the surface damage.

Shear localization is a damage mechanism characterized by the concentration of plastic deformation within thin bands of materials, often observed as a precursor to failure in metals. In this paper, we present a three-dimensional explicit finite element formulation with embedded weak discontinuities for the treatment of shear localization under dynamic loading conditions. In this formulation, the onset of localization is detected via a material stability analysis suitable for rate-sensitive materials, and the continuity of propagating shear bands is ensured using a global tracking strategy involving a heat-conduction type boundary value problem, solved for a scalar level set function over the global domain of the problem. In addition, we propose a modified quadrature rule that is used to compute the contribution of an individual element to the global finite element arrays, taking into account the position of the embedded shear band within that element. The mechanical threshold stress (MTS) model, modified to account for dynamic recrystallization, is adopted as the constitutive model for rate- and temperature-sensitive metallic materials, and the constitutive equations are formulated in a corotational setting. The algorithmic implementation of the integrated finite element framework, including the global tracking strategy, is described in detail. Subsequently, a simple example problem involving shear band formation under uni-axial tension is presented, to illustrate the advantages of the proposed computational framework over conventional finite element techniques, in terms of convergence and mesh insensitivity of their numerical results. In addition, to demonstrate the capabilities of the proposed computational framework in more realistic problems, numerical results are compared to data from split-Hopkinson pressure bar dynamic experiments, conducted on two stainless steel samples with different geometric designs. In each problem presented, we study the effect of the mesh size on the numerical solution, including global results such as the load–displacement response, and local quantities such as material state variables at individual quadrature points. These numerical studies show that the geometry and total volume of the fully-developed embedded shear band are independent of the mesh size.

We present a 3D hybridizable discontinuous Galerkin (HDG) method for nonlinear elasticity which can be efficiently used for thin structures with large deformation . The HDG method is developed for a three-field formulation of nonlinear elasticity and is endowed with a number of attractive features that make it ideally suited for thin structures. Regarding robustness, the method avoids a variety of locking phenomena such as membrane locking, shear locking, and volumetric locking. Regarding accuracy, the method yields optimal convergence for the displacements, which can be further improved by an inexpensive postprocessing. And finally, regarding efficiency, the only globally coupled unknowns are the degrees of freedom of the numerical trace on the interior faces, resulting in substantial savings in computational time and memory storage. This last feature is particularly advantageous for thin structures because the number of interior faces is typically small. In addition, we discuss the implementation of the HDG method with arc-length algorithms for phenomena such as snap-through, where the standard load incrementation algorithm becomes unstable. Numerical results are presented to verify the convergence and demonstrate the performance of the HDG method through simple analytical and popular benchmark problems in the literature.

The reproducing kernel particle methods (RKPM) are meshfree methods arising in mechanics, especially in dealing with problems involving large deformation and singularities. We provide a theoretical analysis of super-convergence in Sobolev norms for reproducing kernel (RK) approximations when the interpolation order p is even. Super-convergence phenomenon means the convergence rate is higher than the order that is generally expected. We distinguish the continuous RK approximation and the discrete RKP approximation. While the continuous RK approximations are proven to be super-convergent when p is even, its discrete counterpart has super-convergence only with uniform particle distribution and special choices of RK kernel functions and support sizes. Moreover, super-convergence does not exist for the discrete RKP approximation with general RK support sizes. The concept of pseudo-super-convergence is then introduced to explain why in practice the super-convergence phenomenon is sometimes observed for general cases although in theory it is not true. Our analysis is general for multi-dimensional RK approximations.

We introduce a fast Fourier spectral method for the multi-species Boltzmann collision operator. The method retains the riveting properties of the single-species fast spectral method (Gamba et al. SIAM J. Sci. Comput., 39 pp. B658--B674 2017) including: (a) spectral accuracy, (b) reduced computational complexity compared to direct spectral method, (c) reduced memory requirement in the precomputation, and (d) applicability to general collision kernels. The fast collision algorithm is then coupled with discontinuous Galerkin discretization in the physical space (Jaiswal et al. J. Comp. Phys., 378 pp. 178--208 2019) to result in a highly accurate deterministic method (DGFS) for the full Boltzmann equation of gas mixtures. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method. Various benchmarks highlighting different collision kernels, different mass ratios, momentum transfer, heat transfer, and in particular the diffusive transport have been studied. The results are directly compared with the direct simulation Monte Carlo (DSMC) method.

The material point method (MPM) has demonstrated itself as an effective numerical method to simulate extreme events with large deformations. However, the original MPM suffers the cell crossing noise because it takes the material points as integration points and employs the piecewise linear grid nodal shape functions whose gradient is discontinuous on the cell boundary. A number of techniques have been developed to alleviate the cell crossing noise. In this paper, a new staggered grid material point method (SGMP) is proposed to eliminate the cell crossing noise very efficiently. The volume integrals in the weak form are evaluated by cell center quadrature instead of particle quadrature as the sum of value of the integrand at each cell center of the background grid multiplied by the corresponding quadrature weight. The physical quantities and the quadrature weights at the cell centers are reconstructed efficiently based on an auxiliary grid, which is obtained by shifting the background grid half the side length of its cell in each direction. Similar to the original MPM, both grids carry no permanent information and can be reset after each time step. In addition, the SGMP evaluates the constitutive equations at the particles, just like the original MPM, to readily model the history-dependent materials. To further reduce the cell crossing noise, a continuous strain rate/vorticity field is established based on the auxiliary grid, whose values are determined by the background grid velocity gradient. The strain rate/vorticity at each particle is interpolated from the auxiliary grid nodal values. Due to the overlap of the cell centers and the corresponding auxiliary grid nodes, a very efficient implementation is established in the SGMP. Numerical studies illustrate that the SGMP is capable of eliminating the cell crossing noise with little extra computational effort and the extra cost ratio reduces as the number of the grid cells or the particles increases.

Vibroacoustic performance of the doubly curved thick shell is explored based on the three dimensional sound propagation approach as well as state space solution. In fact, the main aim is particularly focused on inspecting the influence of using three-dimensional theory through sound transmission loss (STL) of the structure which includes more reliable and accurate results especially for relatively thick and thick shells even in high frequency domain in comparison with other theories. In order to achieve this end, firstly stress and strain components are developed to present the governing equations of thick shell. This procedure is carried out by dividing the shell into s layers. Then, a solution technique is provided on the basis of state vector methodology wherein approximate layer model along with local transfer matrix are performed. Moreover, this method is followed by global transfer matrix method for the all layers of structure. As an outcome, in results section, not only the accuracy of the offered results is proved but also the importance of employing the current theory in high frequencies is revealed. Another remarkable achievement of this work is related to nominate the dip points of STL diagram. On contrary to panels, doubly curved shells contain two dips, because of their both radii of curvatures. In this work, the first dip is nominated as curvature frequency. The second dip is similar to that of panels at high frequency zone. Thus, it is called as coincidence frequency. Finally, the behavior of the transmitted pressure and the effect of curvatures on the position of curvature frequency are discussed.

In this paper, we present an isogeometric analysis (IGA) for phase-field models of three different yet closely related classes of partial differential equations (PDEs): geometric PDEs, high-order PDEs on stationary surfaces, and high-order PDEs on evolving surfaces; the latter can be a coupling of the former two classes. In the context of geometric PDEs, we consider mean curvature flow and Willmore flow problems and their corresponding phase-field approximations which yield second-order and fourth-order nonlinear parabolic PDEs. Through some numerical examples, we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions. Moreover, we study numerically the convergence of these phase-field approximations to the sharp interface solutions . As for the high-order PDEs on stationary surfaces, we consider a model problem which is the Cahn–Hilliard equation on a unit sphere, where the surface is modeled using a diffuse-interface approach. Finally, as a model problem for high-order PDEs on evolving surfaces, we consider a phase-field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface. The model consists of two fourth-order nonlinear PDEs which their direct finite element formulation in a Galerkin framework necessitates smooth basis functions with at least global C 1 continuity; a condition that can be easily satisfied using spline bases in IGA. We solve the coupled equations both in two dimensions, where the vesicle is a curve, and in three dimensions, where the vesicle is a surface. The simulation results agree with the numerical and experimental results from the literature.

A popular approach for dealing with randomness in Topology Optimization (TO) is using a Reliability-Based Topology Optimization (RBTO) routine, where a constraint is placed on the probability of failure. In order to avoid crude Monte Carlo Simulations (MCS), a widely-adopted approximation technique is the First-Order Reliability Method (FORM). Despite being simple in its implementation, FORM necessitates a nested non-linear optimization, which requires considerable computational resources for several stochastic parameters . The contribution of this paper is the development of a novel approach for quantifying multiple material uncertainties in RBTO using random matrix theory (RMT), whereby uncertainty in Young’s modulus, Poisson’s ratio and thickness are all encapsulated within a random stiffness matrix . Two algorithms are introduced, RMT and RMT–FORM, which are compared to MCS and FORM in terms of quality of the derived topology and computational complexity. Two numerical benchmark TO problems are studied, where the objective is to minimize volume, while satisfying a probabilistic compliance constraint. It is shown that RMT–FORM serves as an excellent replacement for FORM, while being considerably more computationally efficient for RBTO problems. The paper also demonstrates how the accuracy and efficiency of a first-order approximation scales with the amount of uncertainty in the problem.

Topology optimization has been widely used in numerous multidisciplinary optimization problems, but reliable industrial manufacturing for optimal results is still severely hampered by problems such as tiny holes or cracks and hinges in structural design. A crucial solution is to consider the minimum scale control directly in the design phase. In this paper, a new method based on the moving morphable component (MMC) is presented. This concept uses a virtual component skeleton (VCS)-based effective connection status (ECS) control approach. The greatest advantage of the proposed method is that the minimum length scale can be determined by the minimum thickness of component, thus fulfilling the precise adjustment of the minimum length scale without the effect of component geometry. The validity of the proposed method is demonstrated for topology optimization of the minimum compliance problem and compliant mechanisms .

In this paper, we consider numerical approximations for solving the anisotropic Cahn–Hilliard model. We combine the Scalar Auxiliary Variable (SAV) approach with the stabilization technique to arrive at a novel Stabilized-SAV approach, where three linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient, are added to enhance the stability while keeping the required accuracy. The schemes are very easy-to-implement and fast in the sense that all nonlinear terms are treated in a semi-explicit way, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Composite structural failure usually entails complicated interactions between intralaminar and interlaminar damage. In this paper, a novel discontinuous solid-shell finite element combined with a 3D enriched cohesive element is formulated for the analysis of matrix cracking and delamination in multilayered composites. The proposed formulations are computationally attractive in modeling thin shell structures while retaining the advantages of solid elements. The bending performance was improved by using the enhanced assumed strain method and the assumed natural strain method, which alleviates potential locking problems. A linear elastic orthotropic law was used for layered shells and cracks therein were modeled with a cohesive zone model. A discrete crack method with floating nodes enriching the developed elements enables a mesh-independent description of ply cracks and their interaction with interface cracks. The applicability of the current formulation was verified with numerical examples including large deformations of thin shell problems and delamination growth in post-buckled laminates. Finally, the explicit and totally discrete modeling of matrix and delamination cracks in low velocity impact of cross-ply and angle-ply laminates was demonstrated. Numerical predictions of structural responses and damage patterns agree well with experimental results.

In this paper, we present a computational homogenization framework to model coupled transient heat conduction with heat generation and chemical kinetics in solids. The proposed method considers that both macro and microscales are under transient heat conduction, being the chemical kinetics (which gives rise to internal heat generation source in the heat conduction problem) defined either in some of the constituents of the microscale or even in the whole microscale. The numerical solution is based on a nested solution strategy, in which the finite element method is used for solving both the macro and the microscale problems, configuring a FE 2 scheme. By solving the coupled problem of transient heat conduction with internal heat generation and chemical kinetics defined in a finite representative volume element , we extract the effective thermal properties and chemical kinetics contribution of the composite and employ them to solve the transient macroscale heat conduction problem (also with internal heat generation). This novel numerical framework is employed in the prediction of the State Of Cure (SOC) in filled elastomers . Numerical solutions of some in-plane heat conduction problems are presented in order to assess the proposed numerical strategy, showing that the multiscale model developed is capable of numerically determining transient non-homogeneous maps of the SOC at the microscale.

In this study, a robust numerical model is developed to guarantee that the C-property for both hydrodynamic and solute transport models and positivity of both water depth and solute concentration . Moreover, special attention is paid to satisfying such properties in the unstructured triangular cells near wet–dry interfaces. In the developed numerical model, the finite-volume central-upwind scheme is adopted to approximate the advective fluxes, and Green’s theorem is applied to estimate the diffusive fluxes . The C-property for hydrodynamic model near wet–dry area is achieved by introducing an auxiliary free surface variable used in the piecewise linear reconstruction and discretizing the bed slope source term in a special form. And the C-property for solute transport is guaranteed by reconstructing secondary variables instead of the primary variables within the frame of central-upwind method, and using the coupling approach for the governing system. The positivity of the solute concentration is satisfied using a proposed diluting time step restriction and the positivity of the water depth is guaranteed by a draining time technique. The desired properties of the proposed numerical model are verified by several numerical experiments.

*The work of Shaw was supported in part by the Engineering and Physical Sciences research council EP/H011072/1. This support is gratefully acknowledged.

We study a family of discontinuous Galerkin methods for the displacement obstacle problem of Kirchhoff plates on two and three dimensional convex polyhedral domains, which are characterized as fourth order elliptic variational inequalities of the first kind. We prove that the error in an H 2 -like energy norm is O ( h α ) for the quadratic method, where α ∈ ( 1 2 , 1 ] is determined by the geometry of the domain. Under additional assumptions on the contact set such that the solution has improved regularity, we derive the optimal error estimate with α ∈ ( 1 , 3 2 ) for the cubic method. Numerical experiments demonstrate the performance of the methods and confirm the theoretical results.

In many mechanical engineering practices, the sample information is usually imprecise due to the complex objective environment or various subjective cognitions. In this study, a kind of inverse problem for identifying the uncertain system parameters with imprecise information is investigated by using the evidence theory. First, the uncertain input parameters to be identified are approximately characterized by evidence variables with subinterval-type focal elements. Through the optimization procedure executed in the given computational model, the output response can be expressed as a group of interval numbers with basic probability assignment (BPA). In the subsequent inverse analysis framework, by cumulating the imprecise experimental response measurements with belief degrees to update the response BPAs, the related interval range of unknown evidence variables can be gradually calibrated toward the true value. To improve the optimization efficiency of output response calculation with respect to various focal elements, a relatively simple metamodel is established as an alternative of the original computational model, where the Legendre-type polynomial and Clenshaw–Curtis point are respectively utilized as the basis function and sample construction strategy. Eventually, numerical results in two examples verify that the uncertain parameter identification can be effectively achieved by the presented method.

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a state-based perydynamic model using two potentials: one associated with hydrostatic strain and the other associated with tensile strain. We show that the dynamic evolution is well-posed in the space of H\"older continuous functions $C^{0,\gamma}$ with H\"older exponent $\gamma \in (0,1]$. Here the length scale of nonlocality is $\epsilon$, the size of time step is $\Delta t$ and the mesh size is $h$. The finite difference approximations are seen to converge to the H\"older solution at the rate $C_t \Delta t + C_s h^\gamma/\epsilon^2$ where the constants $C_t$ and $C_s$ are independent of the discretization. The semi-discrete approximations are found to be stable with time. We present numerical simulations for crack propagation that computationally verify the theoretically predicted convergence rate. We also present numerical simulations for crack propagation in precracked samples subject to a bending load., Comment: 42 pages, 11 figures

The characterization of material properties from image sequences is of paramount importance in assessing structural integrity. A particular application can be found in the field of biomedical engineering, where anatomical structures, whose material properties are frequently sought, can be observed in-vivo through time. A traditional approach for such task, involves the estimation of the displacement field across the image sequence – usually by means of optical flow techniques – and the subsequent estimation of the material properties through the formulation of an inverse mechanical problem. The decoupled nature of this approach leads to cumulative errors across these stages, caused by the lack of consistence between the recovered optical flow and the displacement field delivered by the underlying mechanical model yet to be identified. In recent years, new approaches were proposed to integrate the optical flow and the material identification problems to render a compatible characterization of the displacement field in the sense that they correspond to a mechanical model whose material properties also undergo an identification process. In the present work, a new numerical method for the simultaneous identification of the optical flow field and the material properties of a mechanical model from a sequence of images is presented. Several numerical examples with manufactured solution are solved. Different sources of errors are considered, such as randomized pixel errors and interpolation errors, to analyze the robustness of the approach. The results obtained allow us to conclude that the proposed method performs robustly even in the presence of noise, and is also efficient in terms of the number of iterations required to achieve convergence.

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration , which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method.

Generalized continuum theories become necessary for many scenarios like strain localization phenomena or size effects. In this work, we consider a class of higher order continua, namely micromorphic continua, where the kinematics is enhanced by means of a microstructure undergoing an affine micro deformation. The increasing number of the degrees of freedom in such a theory clearly motivates an application of adaptive methods. For linear elastic micromorphic problems, we have shown the consistency of the resulted dual problem, ensuring an optimal convergence order, in Ju and Mahnken (2017). In this work, we extend the framework of goal-oriented adaptivity to time-dependent problems, i.e. plasticity problems, where a backwards-in-time dual problem is crucial to account for the error accumulation over time caused by both error generation and error transport. Our focus is limited to an adaptive control of spatial discretization errors of the finite element method (FEM), while the temporal discretization errors are not considered for simplicity. Based on duality techniques, we derive exact error representations. For practice, four computable error estimators are proposed, where two different ways to obtain enhanced solutions are considered, and where additionally an approximate forwards-in-time dual problem neglecting error transport is introduced. By means of certain localization techniques, these estimators are used to drive an adaptive mesh refinement algorithm. Their effectiveness is shown by several numerical examples based on a prototype model.

This paper presents a piecewise length scale control method for level set topology optimization . Different from the existing methods, where a unique lower limit or upper limit was applied to the entire design domain, this new method decomposes the topological design into pieces of strip-like components based on the connectivity condition, and then, the lower or upper limit for length scale control could be piecewise and dynamically defined based on each component’s real-time status (such as position, orientation, or dimension). Specifically, a sub-algorithm of structural skeleton identification and segmentation is developed to decompose the structure and its skeleton. Then, a skeleton segment-based length scale control method is developed to achieve the piecewise length scale control effect. In addition, a special type of length scale constrained topology optimization problem that involves an irregular design domain will be addressed, wherein the complex design domain plus the length scale constraint may make the conventional length scale control methods fail to work. Effectiveness of the proposed method will be proved through a few numerical examples.

Volumetric spline parameterization of complex geometry plays a key role in isogeometric analysis (IGA). In this paper, we propose a general framework to construct volumetric parameterization from complex shapes based on directed graph simplification of the l 1 polycube structure. By minimizing the l 1 -norm of the normals on the input triangular meshes, a polycube structure can be generated with robustness, efficiency, and controllability. Then an algorithm is proposed for l 1 polycube structure simplification with a directed graph, which is an abstract structure generated from the polycube. After simplification, the vertex number of the directed graph will decrease, and the polycube structure will become more simple. From the simplified polycube structure, we construct a segmented surface by spline fitting techniques, and finally we fill each block with a trivariate B-spline volume with C 0 -constraints. The proposed method can generate volumetric parameterization without an interior extraordinary vertex, and it has very high potential for constructing volumetric parameterization in IGA simulations. Some volume parameterization examples from complex shapes are presented to show the robustness and efficiency of the proposed method.

This paper proposes a numerical model for the simulation of fracture propagation in brittle materials based on the theory of configurational forces. Configurational forces are adopted as the driving force for fracture in association with a tensile stretch criterion to identify crack tips of propagating cracks . Fracture surfaces are approximated by zero-thickness interface elements embedded into the finite element mesh between the bulk elements in the entire domain. In order to eliminate the mesh bias in regards to the fracture pattern, a new fracture segment is allowed to grow along several interface elements at once. The minimal fracture surface is found using an algorithm based on Graph Theory. The solution strategy is explicit, and the fracture path using this algorithm is identified in a post-processing step . The proposed configurational interface model (CIM) for fracture is validated by the analytical solution for a mode I benchmark and by comparisons with results from a variety of laboratory experiments on cylindrical granite specimens containing one and two initial cracks at different angles, which are characterized by complex curved and partially irregular crack patterns.

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide . The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient . We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.

In this work we review a unified formulation for spectral fuzzy, spectral stochastic, and spectral fuzzy–stochastic FEM. We propose some modifications for the fuzzy and fuzzy–stochastic FEM involving local bases in the non-deterministic dimensions and the incorporation of extended FEM into the non-deterministic FEM. These modifications were previously used for stochastic FEM only. We discuss advantages of the proposed techniques for problems with uncertainties in the geometry and demonstrate their application to computational homogenization of heterogeneous materials with geometrical uncertainties in the microstructure. We address also some theoretical aspects of fuzzy FEM, that have not been tackled in the literature, namely the inner product of fuzzy variables. We address also another important problem, which becomes often the target of criticism of fuzzy approach. The membership function of the fuzzy input has often some degree of arbitrariness: it is known only approximately or it is constructed based on some assumptions. Here we propose a fuzzy FEM representation, which requires only the modal value and support of the fuzzy input parameters and is completely independent of the membership function’s shape. We also discuss the case of imprecise probabilities which allows for severe dimension reduction of the non-deterministic problem if the accurate spectral simulation technique is used. Application of dimension reduction is also demonstrated on the example of computational homogenization.

International audience; In this paper, a highly performing model order reduction technique called Proper Generalized Decomposition (PGD) is applied to the numerical mod-eling of highly transient non-linear thermal phenomena associated with additive manufacturing (AM) powder bed fabrication (PBF) processes. The manufacturing process allows for unprecedented design freedom but fabricated parts often suffer from lower quality mechanical properties associated with the fast transients and high temperature gradients during the localized melting-solidification process. For this reason, an accurate numerical model for the thermal evolutions is a major necessity. This work focuses on providing a low-cost/high accuracy prediction of the high gradient thermal field occurring in a material under the action of a concentrated moving laser source, while accounting for phase changes, material non-linearities and time and space-dependent boundary conditions. An extensive numerical simulation campaign shows that the use of PGD in this context enables a remarkable reduction in the total number of global matrix inversions (5 times less or better) compared to standard techniques when simulating realistic AM PBF scenarios.

Models for fluid-driven fractures in a poroelastic medium involve the simultaneous simulation of multiple physical processes—diffusion of pore pressure, poroelastic stresses, fluid injection into a fractured porous medium, and fracture propagation. To simulate the above-mentioned physics, many algorithms and discretization techniques have been used in the past. The finite volume method has been gaining popularity for simulating solid mechanics problems. The discretization algorithms discussed in the past have used a segregated method for solving the poroelastic momentum balance, where iterations are necessary to obtain a converged solution for each component of the displacement vector. This segregated approach, when coupled with fluid-driven fracture propagation, can lead to excessive simulation times because of the coupled nature of the problem. In this study, an implicit formulation was designed to construct a linear system of equation for the components of the displacement vector to solve the poroelastic momentum balance equation (without iterating between components). Within this formulation, a fluid-driven fracture propagation model was implicitly incorporated using a cohesive zone approach. This methodology provided accurate solutions as shown by the validations in this paper where simulated results were compared with various well-known analytical solutions. The implicitness of the new model allowed significant improvements in computation speed when solving multi-physics problems. The simulation speed-ups were found to be 2 to 8 times when compared with the segregated method.

International audience; Isogeometric shape optimization uses a unique model for the geometric description and for the analysis. The benefits are multiple: in particular, it avoids tedious procedures related to mesh updates. However, although the analysis of complex multipatch structures now becomes tractable with advanced numerical tools, isogeometric shape optimization has not yet been proven to be applicable for designing such structures. Based on the initial concept of integrating design and analysis, we develop a new approach that deals with the shape optimization of non-conforming multipatch structures. The model is built by employing the Free-Form Deformation principle. Introducing NURBS composition drastically simplifies the imposition of the shape updates in case of a non-conforming multipatch configuration. In the case of stiffened structures, the use of embedded surfaces enables to tackle the geometric constraint of connecting interfaces between the panel and the stiffeners during shape modifications. For the analysis, we introduce the embedded Kirchhoff-Love shell formulation. The NURBS composition defines the geometry of the shell while the displacement field is approximated using the same spline functions as for the embedded surface. We also formulate a new mortar method to couple non-conforming Kirchhoff-Love shells which intersect with any angle. We apply the developed method on different examples to demonstrate its efficiency and its potential to optimize complex industrial structures in a smooth manner.

In this paper we consider fully mass-conservative numerical schemes for the simulation of incompressible and immiscible two-phase flow in porous media with capillary pressure. Compared with two kinds of conventional IMplicit Pressure Explicit Saturation (IMPES) schemes, the new schemes deserve a merit that the conservation of mass of both phases can be obtained. The total conservation equation is obtained by the summation of the discretized conservation equation for each phase. This approach is quite different from the conventional IMPES schemes. We present two kinds of fully mass-conservative IMPES schemes to solve the coupled systems for pressure, auxiliary velocity and saturation of each phase. The upwind mixed finite element methods are used to solve the pressure–velocity systems which can be decoupled, and the problems in the decoupled systems can be proved to be well-posed. Moreover, the new schemes are unbiased and the saturation of each phase can be proved to be bounds-preserving if the time step size is smaller than a certain value. The new schemes can also be applied to approximate the incompressible and immiscible two-phase flow in heterogeneous porous media with different capillarity pressures. Several interesting examples of incompressible and immiscible two-phase flow in porous media are presented to demonstrate the robustness of the new algorithms.

It is well known that Galerkin finite element methods suffer from pollution error when solving wave problems. To reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave-like functions so that the exact wavenumber is incorporated into the finite element approximation space. Solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied. Recently, a plane wave enriched isogeometric analysis was introduced for acoustic problems. However, it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other. In this work, we show that isogeometric analysis outperforms finite element method in dealing with pollution. We observe similar behaviour when both the methods are enriched with plane waves. Using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions. However, the latter still leads to smaller errors using similar number of degrees of freedom. In conclusion, we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re-meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour.