Structural Topology Optimization optimizes the mechanical performance of a structure while satisfying some functional constraints. Nearly all approaches proposed in the literature are iterative, and the optimal solution is found by repeatedly solving a Finite Elements Analysis (FEA). It is thus clear that the bottleneck is the high computational effort, as these approaches require solving the FEA a large number of times. In this work, we address the need for reducing the computational time by proposing a reduced basis method that relies on the Functional Principal Component Analysis (FPCA). The methodology has been validated considering a Simulated Annealing approach for the compliance minimization in two classical variable thickness problems. Results show the capability of FPCA to provide good results while reducing the computational times, i.e., the computational time for a FEA analysis is about one order of magnitude lower in the reduced FPCA space.

Granular debris flows compose of coarse solid particles, which may be from disaggregated landslides or well weathered rocks on hill surface. Estimation of agitation and flow process of granular debris flows is of great importance in the prevention of disasters. In this work, we conduct physical experiments of sandpile collapse and impacting three packed wooden blocks. The flow profile, run-out distance and rotation of blocks are measured. To simulate the process, we adopt a material point method (MPM) to model granular flows and a deformable discrete element method (DEM) to model blocks. Each block is treated as comprising nine material points to couple the MPM and DEM, and the acceleration of grid nodes arising from the contacts between granular material and blocks is projected to the discrete element nodes working as body forces. The contacts between blocks are detected using the shrunken point method. The experimental results agree well with the experimental results. Thus, the coupling method of MPM and DEM developed in this work would be helpful in the damage analysis of buildings under impacting from the debris flows.

This paper presents a numerical approach for computing solutions to Biot's fully dynamic model of saturated porous media with incompressible solid and fluid phases. Spatial discretization is based on a three-field (*u*-*w*-*p*) formulation, employing a lowest order Raviart-Thomas mixed element for the fluid Darcy velocity (*w*) and pressure (*p*) fields, and a nodal finite element for skeleton displacement field (*u*). The discretization is constructed based on the natural topology of the variables, and satisfies the LBB stability condition, avoiding locking in the incompressible and undrained limits. Since fluid acceleration is not neglected, the three-field formulation fully captures dynamic behavior even under high frequency loading phenomena. The importance of consistent initial conditions is addressed for poroelasticity equations with the mass-balance constraint, a system of differential algebraic equations. Energy balance is derived for the porous medium, and used to assess accuracy of time integration. To demonstrate the performance of the proposed approach, a variety of numerical studies are carried out including verification with analytical and boundary element solutions and analyses of wave propagation, effect of hydraulic conductivity on damping and frequency content, energy balance, mass lumping, mesh pattern and size, and stability. Some discrepancies found in dynamic poroelasticity results in the literature are also explained.

The model reduction of mechanical problems involving contact remains an important issue in computational solid mechanics. In this article, we propose an extension of the hyper-reduction method based on a reduced integration domain to frictionless contact problems written by a mixed formulation. As the potential contact zone is naturally reduced through the reduced mesh involved in hyper-reduced equations, the dual reduced basis is chosen as the restriction of the dual full-order model basis. We then obtain a hybrid hyper-reduced model combining empirical modes for primal variables with finite element approximation for dual variables. If necessary, the inf-sup condition of this hybrid saddle point problem can be enforced by extending the hybrid approximation to the primal variables. This leads to a hybrid hyper-reduced/full-order model strategy. By this way, a better approximation on the potential contact zone is furthermore obtained. A post-treatment dedicated to the reconstruction of the contact forces on the whole domain is introduced. In order to optimize the offline construction of the primal reduced basis, an efficient error indicator is coupled to a greedy sampling algorithm. The proposed hybrid hyper-reduction strategy is successfully applied to a one-dimensional static obstacle problem with a two-dimensional parameter space and also to a three-dimensional contact problem between two linearly elastic bodies. The numerical results show the efficiency of the reduction technique, especially the good approximation of the contact forces compared to other methods.

Recently the Method of Difference Potentials has been extended to Linear Elastic Fracture Mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the Extended Finite Element Method, were introduced in order to improve the approximation near the crack tip leading to near-optimal convergence rates. Now the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the Fast Fourier Transform is provided with the aim of further increasing the efficiency of the method.

This paper develops a new reliability-based topology optimization framework considering spatially varying geometric uncertainty. Geometric imperfections arising from manufacturing errors are modeled with a random threshold model. The projection threshold is represented by a memoryless transformation of a Gaussian random field, which is then discretized by means of the Expansion Optimal Linear Estimation. The structural response and their sensitivities are evaluated with the polynomial chaos expansion, and the accuracy of the proposed method is verified by Monte Carlo simulations. The performance measure approach is adopted to tackle the reliability constraints in the reliability-based topology optimization problem. The optimized designs obtained with the present method are compared with the deterministic solutions and the reliability-based design considering random variables. Numerical examples demonstrate the efficiency of the proposed method.

An adjoint sensitivity analysis framework to evaluate path-dependent design sensitivities for problems involving inelastic materials and dynamic effects is presented and shown in the context of topology optimization. The overall aim is to present a framework which unifies the sensitivity analyses existing in the literature and provides clear guidelines on how to formulate sensitivity analysis for a wide range of path-dependent system behaviors simulated using finite element analysis (FEA). In particular, the focus in on the identification of proper independent variables for constraint formulation, the overall structure of constraint derivatives arising from discrete FEA equations, and the consistent implementation of sensitivity analysis for diverse problem types. This framework is used to compute sensitivity values for a number of complex problems formulated within FEA for which sensitivity calculations are not available in the literature. The sensitivity values obtained are then rigorously verified using numerical differentiation based on the central difference method. Problem types include the use of enhanced assumed strain elements, plane-stress constraints, nonlocal elastoplastic-damage formulations, kinematic/isotropic hardening, rate dependent materials, finite deformations and dynamics. Finally, topology optimization is carried out using some of the different problem types.

This work introduces a novel, mortar-based coupling scheme for electrode-electrolyte interfaces in three-dimensional finite element models for lithium-ion cells and similar electrochemical systems. The coupling scheme incorporates the widely applied Butler-Volmer charge transfer kinetics, but conceptually also works for other interface equations. Unlike conventional approaches, the coupling scheme allows flexible mesh generation for the electrode and electrolyte phases with non-matching meshes at electrode-electrolyte interfaces. As a result, the desired spatial mesh resolution in each phase and the resulting computational effort can be easily controlled, leading to improved efficiency. All governing equations are solved in a monolithic fashion as a holistic, unified system of linear equations for computational robustness and performance reasons. Consistency and optimal convergence behavior of the coupling scheme are demonstrated in elementary numerical tests, and the discharge of two different, realistic lithium-ion cells, each consisting of an anode, a cathode, and an electrolyte, is also simulated. One of the two cells involves about 1.35 million degrees of freedom and very complex microstructural geometries obtained from X-ray tomography data. For validation purposes, characteristic numerical results from the literature are reproduced, and the coupling scheme is shown to require considerably fewer degrees of freedom than a standard discretization with matching interface meshes to achieve a similar level of accuracy.

]]>A transient finite strain viscoplastic model is implemented in a gradient based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws the implicit Newmark-beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes (MMA). Sensitivities required to form the MMA approximation are derived using the adjoint method. To demonstrate the capability of the algorithm several protective systems are designed in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.

This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution field. The method inherits the main advantage of isogeometric analysis, i.e. preserves the original, exact CAD geometry (for example, given by NURBS), but allows pairing it with an approximation space which is more suitable/flexible for analysis, for example, T-splines, LR-splines, (truncated) hierarchical B-splines, and PHT-splines. This generalization offers the advantage of adaptive local refinement without the need to re-parameterize the domain, and therefore without weakening the link with the CAD model. We demonstrate the use of the method with different choices of the geometry and field spaces, and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for non-nested spaces.

In this paper an approach to blend the Hybrid-Trefftz Finite Element Method (HTFEM) and the Isogeometric Analysis (IGA) called the Isogeometric Trefftz method (IGAT) is presented. The structure of the isogeometric extension of the Trefftz method is formally the same as for its conventional counterpart, except the approximation of the boundary displacements and geometry that are carried out using the Non-Uniform Rational B-Splines (NURBS) instead of polynomials. In other words, only the element boundaries are approximated using NURBS basis while the Trefftz approximation is used in the interior of the elements. For that reason, IGAT can be ranked alongside recently developed Isogeometric Boundary Element Method (IGABEM) the NURBS;-Enhanced Finite Element Method (NEFEM), ,the Isogeometric Local Maximum Entropy method (IGA-LME), and the Isogeometrically enhanced of the Scaled-Boundary Element Method(SBFEM), which all use NURBS approximation at the domain boundary only. Theoretical conjectures made in the paper are accompanied by two examples which show that IGAT leads to excellent results using only a few elements.

Design and analysis of phononic crystals (PnCs) are generally based on the deterministic models without considering effects of uncertainties. However, uncertainties existed in PnCs may have a non-trivial impact on their band structure characteristics. In this paper, a sparse point sampling based Chebyshev polynomial expansion (SPSCPE) method is proposed to estimate the extreme bounds of band structures of PnCs. In the SPSCPE, the interval model is introduced to handle the unknown-but-bounded parameters. Then the sparse point sampling scheme and the finite element method are used to calculate the coefficients of the Chebyshev polynomial expansion. After that, the SPSCPE method is applied for the band structure analysis of PnCs. Meanwhile, the checkerboard and hinge phenomena are eliminating by the hybrid discretization model (HDM). In the end, the genetic algorithm is introduced for topology optimization of phononic crystals with unknown-but-bounded parameters. The specific frequency constraint is considered. Two numerical examples are investigated to demonstrate the effectiveness of the proposed method.

An efficient method for solving large nonlinear problems combines Newton solvers and Domain Decomposition Methods (DDM). In the DDM framework, the boundary conditions can be chosen to be primal, dual or mixed. The mixed approach presents the advantage to be eligible for the research of an optimal interface parameter (often called impedance) which can increase the convergence rate. The optimal value for this parameter is usally too expensive to be computed exactly in practice: an approximate version has to be sought, along with a compromise between efficiency and computational cost. In the context of parallel algorithms for solving nonlinear structural mechanical problems, we propose a new heuristic for the impedance which combines short and long range effects at a low computational cost. This article is protected by copyright. All rights reserved.

Multi-material topology optimization often leads to members containing composite materials. However, in some instances, designers might be interested in using only one material for each member. Therefore, we propose an algorithm that selects a single preferred material from multiple materials per overlapping set. We develop the algorithm, based on the evaluation of both the strain energy and the cross-sectional area of each member, to control the material profile (i.e., number of materials) in each subdomain of the final design. This algorithm actively and iteratively selects materials to ensure a single material is used for each member. In this work, we adopt a multi-material formulation that handles an arbitrary number of volume constraints and candidate materials. To efficiently handle such volume constraints, we employ the Zhang-Paulino-Ramos (ZPR) design variable update scheme for multi-material optimization, which is based upon the separability of the dual objective function of the convex subproblem with respect to Lagrange multipliers. We provide an alternative derivation of this update scheme based on the Karush-Kuhn-Tucker (KKT) conditions. Through numerical examples, we demonstrate that the proposed material selection algorithm, which can be readily implemented in multi-material optimization, along with the ZPR update scheme, is robust and effective for selecting a single preferred material among multiple materials. This article is protected by copyright. All rights reserved.

Simulation-based engineering usually needs the construction of *computational vademecum* to take into account the multiparametric aspect. One example concerns the optimization and inverse identification problems encountered in welding processes. This paper presents a nonintrusive a posteriori strategy for constructing quasi-optimal space-time *computational vademecum* using the higher-order proper generalized decomposition method. Contrary to conventional tensor decomposition methods, based on full grids (eg, parallel factor analysis/higher-order singular value decomposition), the proposed method is adapted to sparse grids, which allows an efficient adaptive sampling in the multidimensional parameter space. In addition, a residual-based accelerator is proposed to accelerate the higher-order proper generalized decomposition procedure for the optimal aspect of *computational vademecum*. Based on a simplified welding model, different examples of *computational vademecum* of dimension up to 6, taking into account both geometry and material parameters, are presented. These *vademecums* lead to real-time parametric solutions and can serve as handbook for engineers to deal with optimization, identification, or other problems related to repetitive task.

Bifurcations of the periodic stationary solutions of nonlinear time-periodic time-delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay-differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map that describes the evolution over the time period. Fold, flip, and Neimark-Sacker bifurcations related to the fixed point of this map are analyzed via center manifold reduction and normal form theorems. The analysis unfolds the approximate stability properties and bifurcations of the stationary solution of the delay-differential equation and, at the same time, allows the approximate computation of the arising period-1, period-2, and quasi-periodic solution branches. The method is demonstrated for the delayed Mathieu-Duffing equation, and the results are verified by numerical continuation.

A temperature-dependent visco-hyperelastic formulation is proposed based on a Eulerian finite element method for large-deformation and multimaterial problems, which is pertinent in the application to pressure-sensitive adhesives. All the basic equations are computed in the Eulerian description because it allows arbitrarily large deformations. This formulation employs Simo's finite-strain viscoelastic model, where hyperelasticity is modeled as a novel strain-energy function of the left Cauchy-Green deformation tensor. The left Cauchy-Green deformation tensor is temporally updated from the Eulerian velocity field. Temperature dependence is described with the time-temperature superposition principle. To validate the proposed approach, we simulated tests of uniaxial tension with various tensile speeds and temperature conditions and performed a steel-ball-drop test on an acrylic pressure-sensitive adhesive.

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First-order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain-driven, stress-driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.

*Hierarchical model reduction* is intended to solve efficiently partial differential equations in domains with a geometrically dominant direction. In many engineering applications, these problems are often reduced to 1-dimensional differential systems. This guarantees computational efficiency yet dumps local accuracy as nonaxial dynamics are dropped. Hierarchical model reduction recovers the secondary components of the dynamics of interest with a combination of different discretization techniques, following up a natural separation of variables. The dominant direction is generally solved by the finite element method or isogeometric analysis to guarantee flexibility, while the transverse components are solved by spectral methods, to guarantee a small number of degrees of freedom. By judiciously selecting the number of transverse modes, the method has been proven to improve significantly the accuracy of purely 1-dimensional solvers, with great computational efficiency. A Cartesian framework has been used so far both in slab domains and cylindrical pipes (including arteries) mapped to Cartesian reference domains. In this paper, we investigate the alternative use of a polar coordinates system for the transverse dynamics in circular or elliptical pipes. This seems a natural choice for applications like computational hemodynamics. In spite of this, the selection of a basis function set for the transverse dynamics is troublesome. As pointed out in the literature—even for simple elliptical problems—there is no “best” basis available. In this paper, we perform an extensive investigation of hierarchical model reduction in polar coordinates to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The nonlocal model treated here is characterized by a double-well potential and is a smooth version of the peridynamic model introduced in the work of Silling. The nonlinear peridynamic evolutions are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale *ε* of nonlocal interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation, the consistency error for the numerical approximation is found to depend on the ratio between mesh size *h* and *ε*. More generally, for local Lagrange interpolation of order *p*≥1, the consistency error is of order *h*^{p}/*ε*. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size *h* is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.

Iterative fast Fourier transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative fast Fourier transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically, some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the 3 iterative schemes investigated overperforms the others for all possible microstructures and all contrasts.

A two-scale model is derived from a fully resolved model where the response of concrete, steel reinforcement, and bond between them are considered. The pertinent “effective” large-scale problem is derived from selective homogenisation in terms of the equilibrium of reinforced concrete considered as a single-phase solid. Variational formulations of the representative volume element problem are established in terms of the subscale displacement fields for the plain concrete continuum and the reinforcement bars. Dirichlet and Neumann boundary conditions (BCs) are imposed on the concrete (pertaining to uniform boundary displacement and constant boundary traction, respectively) and on the reinforcement bars (pertaining to prescribed boundary displacement and vanishing sectional forces, respectively). Different representative volume element sizes and combinations of BCs were used in FE^{2} analyses of a deep beam subjected to four-point bending. Results were compared with those of full resolution (single-scale). The most reliable response was obtained for the case of Dirichlet-Dirichlet BCs, with a good match between the models in terms of the deformed shape, force-deflection relation, and average strain. Even though the maximum crack widths were underestimated, the Dirichlet-Dirichlet combination provided an approximate upper bound on the structural stiffness.

In this paper, we propose a novel approach to combine two complementary numerical models of the dynamical behaviour of mechanical systems: primal (compatible) and dual (equilibrated). This combined model provides an improved estimate of the eigenfrequencies of the system by feeding each model with information from the other. Numerical solutions obtained from both the fundamental models and from the proposed approach are presented and studied. This approach will be applicable to any eigenvalue problem associated with a PDE that can be expressed by complementary numerical models.

Cracks in 3-dimensions (3D) have arbitrary shapes and therefore present difficulties for numerical modelling. A novel adaptive cracking particle method with explicit and accurate description of 3D cracks is described in this paper. In this meshless method, crack surfaces are described by a set of discontinuous segments, which are associated with particles. This group of particles are assumed all to be “cracked” and split into 2 subparticles with modified support domains. Compared to the original method where the spherical supports at particles are equally divided, the proposed method makes use of nonplanar segments to account for changes in crack face direction. The orientations of those segments and the angular changes of cracks during crack propagation steps are recorded using triangular meshes. Supports of weight functions are modified according to those changes so that quasi-continuous crack surfaces can be obtained, avoiding the spurious cracking seen in earlier cracking particle methods. An adaptive approach in 3D is then introduced to capture stress gradients around crack fronts. Several 3D crack problems with reference results have been tested to validate the proposed method with good agreement being achieved using the new method, showing it to be potentially a significant advance for 3D fracture prediction problems.

The boundary-value problems of mechanics can be solved using the material point method with explicit solver formulations. In explicit formulations, even quasi-static problems are solved as if dynamic, which means that waves are reflected at computational boundaries, generating spurious oscillations in the solution to the boundary-value problem. Such oscillations can be reduced to a level such that they are barely noticeable with the use of transmitting boundaries. Current implementations of transmitting boundaries in the material point method are limited to the standard viscous boundary. The absence of any stiffness component in the standard viscous boundary may lead to an undesirable finite rigid-body motion over time. This motion can be minimized through the adoption of the transmitting cone boundary that approximates the stiffness of the unbounded domain. This paper lays out the implementation of the transmitting cone boundary for the generalized interpolation material point method. The cone boundary reflection-canceling tractions can be applied to either the edges or the centroids of material points; this paper discusses the implications of both approaches.

The spatiotemporal response of a stainless steel plate undergoing cyclic laser shock is recorded with an infrared camera, and digital image correlation is used to analyze both displacement and temperature fields. Two very challenging difficulties are addressed: (i) large gray-level variations (due to temperature changes) and (ii) convection effects affecting images. To this aim, a spatiotemporal regularization is designed exploiting a numerical model of the test. The thermomechanical space-time predictions are first processed through Karhunen-Loève decomposition to extract dominant temporal and spatial modes. The temporal modes are then inserted in a spatiotemporal digital image correlation framework to estimate the experimental spatial modes that account for both gray-level variations (and hence temperature) and displacement fields. It is shown that with only 3 modes, the full thermomechanical response of the material is captured. The temporal regularization issued from the model also allows the spurious effect of convection to be filtered out. Due to the drastic decrease in the number of degrees of freedom because of data reduction, the number of analyzed frames can be reduced from 50 down to 6 to capture the thermomechanical response, thereby leading to enhanced efficiency.

In this paper, a mixed spectral element formulation is presented for planar, linear elasticity. The degrees of freedom for the stress are integrated traction components, ie, surface force components. As a result, the tractions between elements are continuous. The formulation is based on minimization of the complementary energy subject to the constraints that the stress field should satisfy equilibrium of forces and moments. The Lagrange multiplier, which enforces equilibrium of forces, is the displacement field and the Lagrange multiplier, which enforces equilibrium of moments, is the rotation. The formulation satisfies equilibrium of forces pointwise if the body forces are piecewise polynomial. Equilibrium of moments is weakly satisfied. Results of the method are given on orthogonal and curvilinear domains, and an example with a point singularity is given.

In this paper, a 3-node *C*^{0} triangular element for the modified couple stress theory is proposed. Unlike the classical continuum theory, the second-order derivative of displacement is included in the weak form of the equilibrium equations. Thus, the first-order derivative of displacement, such as the rotation, should be approximated by a continuous function. In the proposed element, the derivative of the displacement is defined at a node using the node-based smoothed finite element method. The derivative fields, continuous between elements and linear in an element, are approximated with the shape functions in element. Both the displacement field and the derivative field of displacement are expressed in terms of the displacement degree of freedom only. The element stiffness matrix is calculated using the newly defined derivative field. The performance of the proposed element is evaluated through various numerical examples.

The concept of full commutativity of displacements in the expression for strain energy density for the geometrically nonlinear problem has been introduced for the first time and fully established in this paper. Its consequences for the FEM formulation have been demonstrated. As a result, the strain energy, equilibrium equation, and incremental equilibrium equation for the geometrically nonlinear problem can all be presented in a unified manner involving various stiffness matrices that are all symmetric, unique, and explicitly expressed. As an important application, the framework has been employed in the FEM implementation of Koiter's initial postbuckling theory, which has been handicapped by its mesh sensitivity in evaluating one of the initial postbuckling coefficients. This has largely prevented it from being incorporated in mainstream commercial FEM codes. Based on the outcomes of this paper, the mesh sensitivity problem has been completely resolved without the need to use any specially formulated element. As a result, Koiter's theory can be practically and straightforwardly implemented in any FEM code. The results have been verified against those found in the literature.

This paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.

In this paper, we propose a stabilized finite element method for the numerical solution of contact between a small deformation elastic membrane and a rigid obstacle. We limit ourselves to friction-free contact, but the formulation is readily extendable to more complex situations.

In this paper, a high-order finite element method for partial differential equations on smooth surfaces is proposed. The surface is defined as the intersection of a rectangular cuboid and an implicitly defined surface. Therefore, the surface of interest may not be closed. The main novel contribution in this work is the incorporation of an exact geometry description of surfaces with boundary into the finite element method. To this end, a piecewise planar triangulation is mapped onto the surface of interest by making use of the implicit surface definition. The mapping uses predefined search directions and can, therefore, be tailored to consider boundaries. High-order hierarchical shape functions are utilized for the field approximation. They are defined on a reference triangle in the usual way. The proposed method is easy to implement and bypasses the need for a high-order geometry description. Furthermore, due to the exact geometry, the imposition of Dirichlet boundary conditions, source terms, and mesh refinement are easy to carry out.

Two variational principles are proposed that describe equilibrium problems with connected nonlinear beams and solids. The principles extend the classical principle of minimum potential energy for beams and deformable bodies incorporating constraints that weakly enforce the beam kinematics at the common interface using either Lagrange multipliers or penalty terms. In contrast with existing alternatives, in the new approach, the surfaces of bodies connected to beams can deform in an energetically optimal way while globally behaving as beam cross sections. This allows, eg, warping and Poisson effects in beam/solid interfaces. Finite element implementations of the new principles are described in detail and application examples are provided that illustrate their use.

Computation of the distribution of species in hydrocarbon reservoirs from diffusions (thermal, molecular, and pressure) and natural convection is an important step in reservoir initialization. Current methods, which are mainly based on the conventional finite-difference approach, may not be numerically efficient in fractured and other media with complex heterogeneities. In this work, the discontinuous Galerkin (DG) method combined with the mixed finite element (MFE) method is used for the calculation of compositional variation in fractured hydrocarbon reservoirs. The use of unstructured gridding allows efficient computations for fractured media when the cross flow equilibrium concept is invoked. The DG method has less numerical dispersion than the upwind finite-difference methods. The MFE method ensures continuity of fluxes at the interface of the grid elements. We also use the local DG (LDG) method instead of the MFE to calculate the diffusion fluxes. Results from several numerical examples are presented to demonstrate the efficiency, robustness, and accuracy of the model. Various features of convection and diffusion in homogeneous, layered, and fractured media are also discussed.

We describe a heuristic method of triangulating arbitrarily shaped polyhedra without the addition of Steiner points. The polyhedra are simple, with each vertex connected to at least 3 other vertices (ie, coplanarity and colinearity are not considered). They may, however, be convex or concave and consist of dozens or even hundreds of facets. This makes the treatment universal enough to well meet the requirements of models used to simulate fractured rock masses. Certain concepts are defined in the work, eg, adjacent vertices, polygon of adjacent vertices, and closed cone of a vertex. A polygon of adjacent vertices of an apex can be subdivided into a set of nonoverlapping triangles without adding any vertices. These triangles, together with the apex, form tetrahedra whose union is the closed cone of the apex. The polyhedron is thus the union of the closed cones. Subsequently, we triangulate the polyhedron by gradually removing the closed cones of its vertices. The number of vertices of the polyhedron decreases by one each time a closed cone is removed. A block with *n* vertices can produce no more than *n*−3 tetrahedra. We present the analysis procedure and discuss the core issues of the method proposed.

The isogeometric approach to computational engineering analysis makes use of nonuniform rational B-splines to discretise both the geometry and the analysis field variables, giving a higher-fidelity geometric description and leading to improved convergence properties of the solution over conventional piecewise polynomial descriptions. Because of its boundary-only modelling, with no requirement for a volumetric nonuniform rational B-spline geometric definition, the boundary element method is an ideal choice for isogeometric analysis of solids in 3D. An isogeometric boundary element analysis (IGABEM) algorithm is presented for the solution of such problems in elasticity and is accelerated using the black-box fast multipole method (bbFMM). The bbFMM scheme is of *O*(*n*) complexity, giving a general kernel-independent separation that can be easily integrated into existing conventional IGABEM codes with little modification. In the bbFMM scheme, an important process of obtaining a low-rank approximation of moment-to-local operators has been hitherto based on singular value decomposition, which can be very time consuming for large 3D problems, and this motivates the present work. We introduce the proper generalized decomposition method as an alternative approach, and this is demonstrated to enhance efficiency in comparison with schemes that rely on the singular value decomposition. In the worst case, a factor of approximately 2 performance gain is achieved. Numerical examples show the performance gains that are achievable in comparison to standard IGABEM solutions and demonstrate that solution accuracy is not affected. The results illustrate the potential of this numerical technique for solving arbitrary large scale elastostatics problems directly from computer-aided design models.

We present an original mathematical formulation for optimizing structural topology while simultaneously identifying an optimal set of design materials that are selected from a larger set of candidate materials. This design task is analogous to that, which is commonly encountered in additive manufacturing applications in which the 3D printer can print parts containing up to 3 distinct materials that can be selected from a larger suite of usable materials. The material distribution is parameterized via the shape functions with penalization formulation in which a set of activation functions, which are derived from a partition of the unit hypercube, is used to determine the effective local elasticity modulus within a single finite element. Additionally, we introduce an inverse *p*-norm function, which is used to ensure that the optimized material properties converge to a set of discrete values corresponding to the available candidate materials. The algorithm has been implemented on a set of 2D benchmark problems. Numerical results show that the formulation combining the inverse *p*-norm function with the activation functions successfully produces optimized multimaterial solutions containing no more than the prescribed number of distinct materials.

A robust mesh optimisation method is presented that directly enforces the resulting deformation to be orientation preserving. Motivated by aspects from mathematical elasticity, the energy functional of the mesh deformation can be related to a stored-energy functional of a hyperelastic material. Formulating the functional in the principal invariants of the deformation gradient allows fine-grained control over the resulting deformation. Solution techniques for the arising nonconvex and highly nonlinear system are presented. As existing preconditioners are not sufficient, a partial differential equation–based preconditioner is developed.

A recent unsymmetric 4-node, 8-DOF plane element US-ATFQ4, which exhibits excellent precision and distortion-resistance for linear elastic problems, is extended to geometric nonlinear analysis. Since the original linear element US-ATFQ4 contains the analytical solutions for plane pure bending, how to modify such formulae into incremental forms for nonlinear applications and design an appropriate updated algorithm become the key of the whole job. First, the analytical trial functions should be updated at each iterative step in the framework of updated Lagrangian formulation that takes the configuration at the beginning of an incremental step as the reference configuration during that step. Second, an appropriate stress update algorithm in which the Cauchy stresses are updated by the Hughes-Winget method is adopted to estimate current stress fields. Numerical examples show that the new nonlinear element US-ATFQ4 also possesses amazing performance for geometric nonlinear analysis, no matter whether regular or distorted meshes are used. It again demonstrates the advantages of the unsymmetric finite element method with analytical trial functions.

We present a modification of the multiscale finite element method (MsFEM) for modeling of heterogeneous viscoelastic materials and an enhancement of this method by the adaptive generation of both meshes, ie, a macroscale coarse one and a microscale fine one. The fine mesh refinements are performed independently within coarse elements adjusting the microscale discretization to the microstructure, whereas the coarse mesh adaptation optimizes the macroscale approximation. Besides the coupling of the *hp*-adaptive finite element method with the MsFEM we propose a modification of the MsFEM to accommodate for the analysis of transient nonlinear problems. We illustrate the efficiency and accuracy of the new approach for a number of benchmark examples, including the modeling of functionally graded material, and demonstrate the potential of our improvement for upscaling nonperiodic and nonlinear composites.

We investigate spatial stability with various numerical discretizations in displacement and pressure fields for poroelasticity. We study 2 sources of the early time instability: discontinuity of pressure and violation of the inf-sup condition. We consider both compressible and incompressible fluids by employing the monolithic, stabilized monolithic, and fixed-stress sequential methods. Four different spatial discretization schemes are used: Q1Q1, Q2Q1, Q1P0, and Q2P0. From mathematic analysis and numerical tests, the piecewise constant finite volume method for flow provides stability at the early time for the case of the pressure discontinuity. On the other hand, a piecewise continuous (or higher-order) interpolation of pressure shows spatial oscillation, having lower limits of time step size, although lower approximations of pressure than displacement can alleviate the oscillation. For an incompressible fluid, Q2Q1 can be better than Q1P0, because Q1P0 might not satisfy the inf-sup condition. However, regardless of fluid compressibility and the pressure discontinuity, the fixed-stress method can effectively stabilize the oscillation without an artificial stabilizer. Even when Q1P0 and Q1Q1 with the monolithic method cannot satisfy the inf-sup condition, the fixed-stress method can yield the full-rank linear system, providing stability. Thus, the fixed-stress method with Q1P0 can effectively circumvent the aforementioned 2 types of instability.

We present a variational method for problems in solid and structural mechanics that is designed to be intrinsically free from locking when using equal-order interpolation for all involved fields. The specific feature of the formulation is that it avoids all geometrical locking effects (as opposed to material locking effects, for instance Poisson locking) for any type of structural or solid model, independent of the underlying discretization scheme. The possibility of employing equal-order interpolation for all involved fields circumvents the task of finding particular function spaces to remove locking and avoid artificial stress oscillations. This is particularly attractive, for instance, for isogeometric analysis using unstructured meshes or T-splines. Comprehensive numerical tests underline the promising behavior of the proposed method for geometrically linear and nonlinear problems in terms of displacements and stress resultants using standard finite elements, isogeometric finite elements, and a meshless method.

Gaussian process (GP) metamodels have been widely used as surrogates for computer simulations or physical experiments. The heart of GP modeling lies in optimizing the log-likelihood function with respect to the hyperparameters to fit the model to a set of observations. The complexity of the log-likelihood function, computational expense, and numerical instabilities challenge this process. These issues limit the applicability of GP models more when the size of the training data set and/or problem dimensionality increase. To address these issues, we develop a novel approach for fitting GP models that significantly improves computational expense and prediction accuracy. Our approach leverages the smoothing effect of the nugget parameter on the log-likelihood profile to track the evolution of the optimal hyperparameter estimates as the nugget parameter is adaptively varied. The new approach is implemented in the **R** package **GPM** and compared to a popular GP modeling **R** package (**GPfit)** for a set of benchmark problems. The effectiveness of the approach is also demonstrated using an engineering problem to learn the constitutive law of a hyperelastic composite where the required level of accuracy in estimating the response gradient necessitates a large training data set.

This work presents a new original formulation of the discrete element method (DEM) with deformable cylindrical particles. Uniform stress and strain fields are assumed to be induced in the particles under the action of contact forces. Particle deformation obtained by strain integration is taken into account in the evaluation of interparticle contact forces. The deformability of a particle yields a nonlocal contact model, it leads to the formation of new contacts, it changes the distribution of contact forces in the particle assembly, and it affects the macroscopic response of the particulate material. A numerical algorithm for the deformable DEM (DDEM) has been developed and implemented in the DEM program DEMPack. The new formulation implies only small modifications of the standard DEM algorithm. The DDEM algorithm has been verified on simple examples of an unconfined uniaxial compression of a rectangular specimen discretized with regularly spaced equal bonded particles and a square specimen represented with an irregular configuration of nonuniform-sized bonded particles. The numerical results have been verified by a comparison with equivalent finite element method results and available analytical solutions. The micro-macro relationships for elastic parameters have been obtained. The results have proved to have enhanced the modeling capabilities of the DDEM with respect to the standard DEM.

This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so-called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle-point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact-specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.

A novel integration scheme is proposed for fictitious domain finite element methods. It relies on the use of a surface tracking strategy based on anisotropic mesh adaptation. Thanks to an error estimator, the method builds iteratively an adapted anisotropic mesh that is refined near the geometrical interface and elongated in the direction of a small curvature. This strategy allows to decrease the integration cost, which can be problematic for high-order fictitious domain methods. In addition, it opens the possibility for the creation of unfitted solid shell strategies that can be used for the treatment of thin structures. Numerical studies show that the method leads to promising results for both integration cost and behavior with respect to locking.

In this paper, we present a general model for non-Fickian diffusion and drug dissolution from a controlled drug delivery device coated with a thin polymeric layer. First, we study the stability and deduce an analytic solution to the problem. Then, we consider this solution and provide suitable boundary conditions to replace the problem of mass transport in the coating of a coronary drug-eluting stent. With this approach, we reduced the computational cost of performing numerical simulations in complex 3-dimensional geometries. The model for mass transport by a coronary drug-eluting stent is coupled with a non-Newtonian blood model flow. In order to show the effectiveness of the method, numerical experiments and a model validation with experimental data are also included. In particular, we investigate the influence of the non-Newtonian flow regime on the drug deposition in the arterial wall.

By exploiting the meshless property of kernel-based collocation methods, we propose a fully automatic numerical recipe for solving interpolation/regression and boundary value problems adaptively. The proposed algorithm is built upon a least squares collocation formulation on some quasi-random point sets with low discrepancy. A novel strategy is proposed to ensure that the fill distances of data points in the domain and on the boundary are in the same order of magnitude. To circumvent the potential problem of ill-conditioning due to extremely small separation distance in the point sets, we add an extra dimension to the data points for generating shape parameters such that nearby kernels are of distinctive shape. This effectively eliminates the needs of shape parameter identification. Resulting linear systems were then solved by a greedy trial space algorithm to improve the robustness of the algorithm. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed methods.

In this paper, by combining the dimension splitting method and the improved complex variable element-free Galerkin method, the dimension splitting and improved complex variable element-free Galerkin (DS-ICVEFG) method is presented for 3-dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2-dimensional ones, which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method for each 2-dimensional problem, the improved complex variable moving least-square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1-dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS-ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.

A new concept for hybrid discontinuous Galerkin (DG) methods is presented: control points. These are defined on the interelement boundaries. The concept makes it possible to formulate element shape functions without nodes. Moreover, the theory is not restricted to certain element shapes. Furthermore, one can formulate the discrete model such that the displacement is either continuous or discontinuous at the control points. Classical continuous isoparametric elements are included as special case. As an additional new feature, a regularization technique for very high strain rate sensitivity exponents up to 1000 in finite single crystal viscoplasticity is presented and implemented into the new hybrid DG framework. In addition, the numerical linearization used in an earlier work is carried out analytically in this work. To the knowledge of the authors, this work presents the first hybrid DG implementation of geometrically nonlinear plasticity, here in the context of single crystal plasticity. The regularization method in combination with the DG formulations facilitates a very simple implementation leading to a numerically efficient, robust, and locking-free model. Two examples are investigated: the deformation of a planar double slip single crystal exhibiting localization in the form of shear bands and an oligocrystal under uniaxial load.

In this paper, we propose an approach for reliability-based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability-based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability-based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level-set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.

In this paper, a new effective boundary node method is presented for the solution of acoustic problems, directly in time domain, using exponential basis functions. Unlike many other methods using boundary information, the final coefficient matrix is sparse. The formulation is well suited for domains whose extent is relatively larger than the distance traveled by the acoustic wave in an increment of time. The exponential basis functions used satisfy the time-space governing equation. This helps to choose a relatively large time increment and a moderate number of boundary points, which leads to reduction of computation time. The computation is performed incrementally using a weighted residual in time. Through a series of numerical examples, it is shown that the method, when combined with a domain decomposition strategy, is effectively capable of solving various 1- to 3-dimensional acoustic problems.

A parallel multigrid (MG) method is developed to reduce the large computational costs involved by the finite element simulation of highly viscous fluid flows, especially those resulting from metal forming applications, which are characterized by using a mixed velocity/pressure implicit formulation, unstructured meshes of tetrahedra, and frequent remeshings. The developed MG method follows a hybrid approach where the different levels of nonnested meshes are geometrically constructed by mesh coarsening, while the linear systems of the intermediate levels result from the Galerkin algebraic approach. A linear *O*(*N*) convergence rate is expected (with *N* being the number of unknowns), while keeping software parallel efficiency. These objectives lead to selecting unusual MG smoothers (iterative solvers) for the upper grid levels and to developing parallel mesh coarsening algorithms along with parallel transfer operators between the different levels of partitioned meshes. Within the utilized PETSc library, the developed MG method is employed as a preconditioner for the usual conjugate residual algorithm because of the symmetric undefinite matrix of the system to solve. It shows a convergence rate close to optimal, an excellent parallel efficiency, and the ability to handle the complex forming problems encountered in 3-dimensional hot forging, which involve large material deformations and frequent remeshings.

An approach for investigating finite deformation contact problems with frictional effects with a special emphasis on nonsmooth geometries such as sharp corners and edges is proposed in this contribution. The contact conditions are separately enforced for point contact, line contact, and surface contact by employing 3 different sets of Lagrange multipliers and, as far as possible, a variationally consistent discretization approach based on mortar finite element methods. The discrete unknowns due to the Lagrange multiplier approach are eliminated from the system of equations by employing so-called dual or biorthogonal shape functions. For the combined algorithm, no transition parameters are required, and the decision between point contact, line contact, and surface contact is implicitly made by the variationally consistent framework. The algorithm is supported by a penalty regularization for the special scenario of nonparallel edge-to-edge contact. The robustness and applicability of the proposed algorithms are demonstrated with several numerical examples.

We extend Locally Refined (LR) B-splines to LR T-splines within the Bézier extraction framework. This discretization technique combines the advantages of T-splines to model the geometry of engineering objects exactly with the ability to flexibly carry out local mesh refinement. In contrast to LR B-splines, LR T-splines take a T-mesh as input instead of a tensor-product mesh. The LR T-mesh is defined, and examples are given how to construct it from an initial T-mesh by repeated meshline insertions. The properties of LR T-splines are investigated by exploiting the Bézier extraction operator, including the nested nature, linear independence, and the partition of unity property. A technique is presented to remove possible linear dependencies between LR T-splines. Like for other spline technologies, the Bézier extraction framework enables to fully use existing finite element data structures.

The concept of energy-sampling stabilization is used to develop a mean-strain quadratic 10-node tetrahedral element for the solution of geometrically nonlinear solid mechanics problems. The development parallels recent developments of a “composite” uniform-strain 10-node tetrahedron for applications to linear elasticity and nonlinear deformation. The technique relies on stabilization by energy sampling with a mean-strain quadrature and proposes to choose the stabilization parameters as a quasi-optimal solution to a set of linear elastic benchmark problems. The accuracy and convergence characteristics of the present formulation are tested on linear and nonlinear benchmarks and compare favorably with the capabilities of other mean-strain and high-performance tetrahedral and hexahedral elements for solids, thin-walled structures (shells), and nearly incompressible structures.

We propose a novel finite-element method for polygonal meshes. The resulting scheme is *h**p*-adaptive, where *h* and *p* are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis-aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices.

With the postulation of the principle of virtual action, we propose, in this paper, a variational framework for describing the dynamics of finite dimensional mechanical systems, which contain frictional contact interactions. Together with the contact and impact laws formulated as normal cone inclusions, the principle of virtual action directly leads to the measure differential inclusions commonly used in the dynamics of nonsmooth mechanical systems. The discretization of the principle of virtual action in its strong and weak variational form by local finite elements in time provides a structured way to derive various time-stepping schemes. The constitutive laws for the impulsive and nonimpulsive contact forces, ie, the contact and impact laws, are treated on velocity-level by using a discrete contact law for the percussion increments in the sense of Moreau. Using linear shape functions and different quadrature rules, we obtain three different stepping schemes. Besides the well-established Moreau time-stepping scheme, we can present two alternative integrators referred to as symmetric and variational Moreau-type stepping schemes. A suitable benchmark example shows the superiority of the newly proposed integrators in terms of energy conservation properties, accuracy, and convergence.

In this paper, we extend to Reissner-Mindlin plate bending problems a technique, originally proposed in the context of two-dimensional and three-dimensional continua, for recovering fully equilibrated stresses from the solution of a compatible finite element model. The technique involves partition of unity functions and the analyses of overlapping star patches modelled with hybrid equilibrium plate elements. The patches are subjected to balanced systems of loads composed of partitioned and fictitious loads, where the latter are derived from the stresses of the compatible solution. The special case of assumed linear displacement fields of both deflection and rotation for the compatible model is included. This case requires additional fields of stress resultants to correct possible rotational imbalances of star patches, and these are derived elementwise. Other cases of nonconforming elements are briefly considered. Numerical examples are presented to illustrate the effectiveness of these techniques in terms of the deviation of the recovery, which compares the complementary strain energy of a recovered solution with that obtained by a global equilibrated analysis based on the same stress approximations.

In this paper, three new kinds of time-domain numerical methods of exponentially damped systems are presented, namely, the simplified Newmark integration method, the precise integration method, and the simplified complex mode superposition method. Based on the traditional Newmark integration method and transforming the equation of motion with exponentially damping kernel functions into an equivalent second-order equation of motion by using the internal variables technique, the simplified Newmark integration method is developed by using a decoupling technique to reduce the computer run time and storage. By transforming the equation of motion with exponentially damping kernel functions into a first-order state-space equation, the precise integration technique is used to numerically solve the state-space equation. Based on a symmetric state-space equation and the complex mode superposition method, a delicate and simplified general solution of exponentially damped linear systems, completely in real-value form, is developed. The accuracy and efficiency of the developed numerical methods are compared and discussed by two benchmark examples.

Implicit gradient plasticity models incorporate higher-order spatial gradients via an additional Helmholtz type equation for the plastic multiplier. So far, the enrichment has been limited to second-order spatial gradients, resulting in a formulation that can be discretised using -continuous finite elements. Herein, an implicit gradient plasticity model is formulated that includes a fourth-order gradient term as well. A comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second-order gradient plasticity model is made using a dispersion analysis. The higher-order continuity requirement for the fourth-order implicit gradient plasticity model has been met by exploiting the higher-order continuity property of isogeometric analysis, which uses nonuniform rational B-splines as shape functions instead of Lagrange polynomials. The discretised variables, displacements, and plastic multiplier may require different orders of interpolation, an issue that is also addressed. Numerical results show that both formulations can be used as a localisation limiter, but that quantitative differences occur, and a different evolution of the localisation band is obtained for 2-dimensional problems.

In this work, a new strategy for solving multiscale topology optimization problems is presented. An alternate direction algorithm and a precomputed offline microstructure database (Computational Vademecum) are used to efficiently solve the problem. In addition, the influence of considering manufacturable constraints is examined. Then, the strategy is extended to solve the coupled problem of designing both the macroscopic and microscopic topologies. Full details of the algorithms and numerical examples to validate the methodology are provided.

No abstract is available for this article.

]]>The Chang-Hicher micromechanical model based on a static hypothesis, not unlike other models developed separately at around the same epoch, has proved its efficiency in predicting soil behaviour. For solving boundary value problems, the model has now integrated stress-strain relationships by considering both the micro and macro levels. The first step was to solve the linearized mixed control constraints by the introduction of a predictor–corrector scheme and then to implement the micro–macro relationships through an iterative procedure. Two return mapping schemes, consisting of the closest-point projection method and the cutting plane algorithm, were subsequently integrated into the interparticle force-displacement relations. Both algorithms have proved to be efficient in studies devoted to elementary tests and boundary value problems. Closest-point projection method compared with cutting plane algorithm, however, has the advantage of being more intensive cost efficient and just as accurate in the computational task of integrating the local laws into the micromechanical model. The results obtained demonstrate that the proposed linearized method is capable of performing loadings under stress and strain control. Finally, by applying a finite element analysis with a biaxial test and a square footing, it can be recognized that the Chang-Hicher micromechanical model performs efficiently in multiscale modelling.

Single-curvature plates are commonly encountered in mechanical and civil structures. In this paper, we introduce a topology optimization method for the stiffness-based design of structures made of curved plates with fixed thickness. The geometry of each curved plate is analytically and explicitly represented by its location, orientation, dimension, and curvature radius, and therefore, our method renders designs that are distinctly made of curved plates. To perform the primal and sensitivity analyses, we use the geometry projection method, which smoothly maps the analytical geometry of the curved plates onto a continuous density field defined over a fixed uniform finite element grid. A size variable is ascribed to each plate and penalized in the spirit of solid isotropic material with penalization, which allows the optimizer to remove a plate from the design. We also introduce in our method a constraint that ensures that no portion of a plate lies outside the design envelope. This prevents designs that would otherwise require cuts to the plates that may be very difficult to manufacture. We present numerical examples to demonstrate the validity and applicability of the proposed method.

In this paper, a novel characteristic–based penalty (CBP) scheme for the finite-element method (FEM) is proposed to solve 2-dimensional incompressible laminar flow. This new CBP scheme employs the characteristic-Galerkin method to stabilize the convective oscillation. To mitigate the incompressible constraint, the selective reduced integration (SRI) and the recently proposed selective node–based smoothed FEM (SNS-FEM) are used for the 4-node quadrilateral element (CBP-Q4SRI) and the 3-node triangular element (CBP-T3SNS), respectively. Meanwhile, the reduced integration (RI) for Q4 element (CBP-Q4RI) and NS-FEM for T3 element (CBP-T3NS) with CBP scheme are also investigated. The quasi-implicit CBP scheme is applied to allow a large time step for sufficient large penalty parameters. Due to the absences of pressure degree of freedoms, the quasi-implicit CBP-FEM has higher efficiency than quasi-implicit CBS-FEM. In this paper, the CBP-Q4SRI has been verified and validated with high accuracy, stability, and fast convergence. Unexpectedly, CBP-Q4RI is of no instability, high accuracy, and even slightly faster convergence than CBP-Q4SRI. For unstructured T3 elements, CBP-T3SNS also shows high accuracy and good convergence but with pressure oscillation using a large penalty parameter; CBP-T3NS produces oscillated wrong velocity and pressure results. In addition, the applicable ranges of penalty parameter for different proposed methods have been investigated.

A computational certification framework under limited experimental data is developed. By this approach, a high-fidelity model (HFM) is first calibrated to limited experimental data. Subsequently, the HFM is employed to train a low-fidelity model (LFM). Finally, the calibrated LFM is utilized for component analysis. The rational for utilizing HFM in the initial stage stems from the fact that constitutive laws of individual microphases in HFM are rather simple so that the number of material parameters that needs to be identified is less than in the LFM. The added complexity of material models in LFM is necessary to compensate for simplified kinematical assumptions made in LFM and for smearing discrete defect structure. The first-order computational homogenization model, which resolves microstructural details including the structure of defects, is selected as the HFM, whereas the reduced-order homogenization is selected as the LFM. Certification framework illustration, verification, and validation are conducted for ceramic matrix composite material system comprised of the 8-harness weave architecture. *Blind* validation is performed against experimental data to validate the proposed computational certification framework.

A key limitation of the most constitutive models that reproduce a degradation of quasi-brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations. The essential constituent of the fatigue model is an implicit gradient-enhanced formulation of the damage rate. The evolution of the field variables is computed as a multiscale Fourier series in time. On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarser macrochronological scale associated with the duration of material deterioration. The performance of the developed method is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.