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International audience; This work aims at proposing a new procedure for parametric problems whose separated representation has been considered difficult, or whose SVD compression impacted the results in terms of performance and accuracy. The proposed technique achieves a fully separated representation for layered domains with interfaces exhibiting waviness or – more generally – deviating from planar surfaces, parallel to the coordinate plane. This will make possible a simple separated representation, equivalent to others, already analyzed in some of our former works. To prove the potentialities of the proposed approach, two benchmarks will be addressed, one of them involving an efficient space–time separated representation achieved by considering the same rationale.

A block coordinate descent algorithm within a discontinuous Galerkin finite element formulation is proposed for both explicit and implicit energy minimization in solid bodies in which rigid-cohesive fractures initiate and propagate. The crack opening dependent Barenblatt type surface cohesive energy function is non-differentiable at the origin. In each iteration or each explicit update, the algorithm minimizes the potential with respect to each crack opening displacement unknown and with respect to the block of deformation unknowns, sequentially. This decomposes minimization of the full non-differentiable problem into a number of “small” non-differentiable sub-problems that must be solved locally at the inter-element boundaries of the finite element mesh and an “easy” differentiable sub-problem that characterizes global equilibrium. As a result, non-differentiability is effectively treated locally and the algorithm can be easily incorporated into standard finite element codes. On the basis of a convexity analysis of the proposed non-differentiable energy functional, we obtain a minimum cohesive process zone resolution criterion, known empirically in the previous literature as a requirement for capturing the amount of dissipated fracture energy correctly. The method is free of any regularization parameters and preserves the discrete nature of fracture without introducing a stress discontinuity at initiation of decohesion. Robustness of the method is shown through several numerical examples of fragmentation and branching and through comparisons with existing numerical and experimental results.

The variational phase-field method is an attractive non-local approach of modeling fracture in heterogeneous materials. However, these materials usually require a fine mesh to resolve the fracture process zone. Consequently, the standard finite element solver becomes cumbersome due to the large number of elements in applications with highly heterogeneous materials. Motivated by this limitation, an algorithm based on FFT methods has been introduced in this paper to solve the phase-field model of brittle fracture . Relying on a staggered update scheme, the proposed algorithm solves the fracture problem and mechanical problem separately, both using the FFT technique. It inherits the advantages of classical FFT methods in terms of simplicity of mesh generation and parallel implementation. Introduced within a FFT-based code “AMITEX”, it takes the advantage of massively parallel capabilities associated with a distributed memory implementation. The characteristics of the proposed method are analyzed in a single edge notched specimen benchmark. Representative numerical examples demonstrate that the proposed FFT solver is capable of predicting different crack modes and complex crack configuration , such as crack interaction, branching and coalescence . Finally, a model of an idealized continuous fiber composite with void involving over 32 million voxels is solved, illustrating the potential of the FFT solver in large-scale problems.

We propose a parallel and computationally efficient multi-split XFEM approach for 3-D analysis of heterogeneous materials. In this approach, multiple discontinuities (pores and reinforcement particles) may intersect any given element (we call those elements multi-split elements). These discontinuities are modeled by imposing additional degrees of freedom at the nodes. The main advantage of the proposed scheme is that the mesh size remains independent of the relative distance among the heterogeneities/discontinuities. The pores and reinforcement particles are assumed to be spherical. The simulations are performed for uniform and non-uniform heterogeneity distribution. The Young’s modulus of the heterogeneous material is evaluated for different amount of pores and reinforcement particles. To demonstrate the computational efficiency of the multi-split XFEM, elastic damage analysis is performed for the unit cell with 5% pores and 5% reinforcement particles under uniaxial tensile loading. These simulations show that the Young’s modulus decreases linearly with the increase in the volume fraction of the pores and increases linearly with the increase in volume fraction of reinforcement particles. The multi-split XFEM is found to be at least 1.8 times computationally efficient than standard XFEM and at least 6.7 times computationally efficient than FEM.

The model validation with respect to epistemic uncertainty , where only a small amount of experimental data is available, is a challenging problem in practical engineering. Interval theory is a useful tool to deal with such epistemic uncertainty, and this paper aims to construct an interval theory-based analysis framework for model validation. Using the statistical moments of experimental observations, an unbiased estimation method is firstly presented to quantify the interval bounds of system uncertainties. In the subsequent process of predicting uncertain responses, the Legendre polynomial chaos expansion is introduced as the surrogate model , which can greatly improve the computational efficiency of uncertainty propagation . Then the concept of interval fitting degree is proposed to establish a new quantitative validation metric, which can accurately characterize the agreement between the computational response interval and experimental response interval. Meanwhile, an operation of interval parameter calibration is executed in the form of optimization to improve the prediction accuracy of computational model. Finally, the Sandia thermal challenge problem is utilized to verify the feasibility of presented model validation method in engineering application.

This paper presents an isogeometric analysis (IGA) based design method to address the stress-constrained topology optimization problem of plane stress and bending of thin plates. Based on the popular SIMP model, the integrated framework of geometry modeling, structural stress analysis and optimization is established. Owing to the geometry exactness and high-order continuity between elements, the IGA improves the computational accuracy of stress, and thus enhances the credibility of optimum design. Meanwhile, the obvious zigzag boundaries are avoided in the optimized results, and the stress function of IGA maintains the continuity for a relatively coarse discretization . Moreover, the IGA-SIMP method can easily meet the requirement of C 1 -continuity for the Kirchhoff plate formulations, which also facilitates the stress analysis and sensitivity calculation. To overcome the convergence difficulty of highly nonlinear stress aggregation constraint, two STM (stability transformation method)-based stabilization schemes combining with the P-norm function for global stress constraint are developed to achieve the stable iterations and acceptable designs. Finally, representative examples illustrate the effectiveness and convenience of the proposed approach. It is indicated that the IGA-SIMP method shows superior performance for solution accuracy and efficiency, and the local stress level is well controlled.

Most of the existing methods of brake squeal instability analysis are merely available to handle single type of uncertain case. In this study, an improved unified method is developed for uncertainty quantification, which is capable of handling two types of fuzzy–interval cases. In the first fuzzy–interval case, uncertain parameters of engineering structures are assumed as either fuzzy variables or interval variables, which exist in structures simultaneously and independently. In the second fuzzy–interval case, all uncertain parameters are represented by interval variables, but their lower and upper bounds just can be expressed as fuzzy variables instead of deterministic values. In the proposed method, fuzzy–boundary interval variables are introduced to handle fuzzy–interval uncertainties, and based on which an improved response analysis model is established. In the improved model, the fuzzy–boundary interval variables are firstly converted into interval-boundary variables by level-cut technique. Then by temporarily neglecting boundary uncertainties, the initial interval responses can be approximated via conducting once Taylor series expansion and subinterval analysis. Next, Taylor series expansion and central difference method are combined twice to deal with boundary uncertainties, and the interval responses of the structures with interval-boundary variables are yielded. Finally, the fuzzy–interval responses of the structures are derived on the basis of interval union operation and fuzzy decomposition theorem. The improved method is subsequently extended to quantify the uncertainties in brake squeal instability analysis involving two types of fuzzy–interval uncertainties. The effectiveness of the proposed method on tackling fuzzy–interval problems is demonstrated by numerical examples.

This paper presents a topology optimization method for structures made of multiple composite components (substructures) with tailored material orientations. It is capable of simultaneously optimizing the overall topology, component partitioning, and material orientation for each component, via a vector field variable that specifies fractional membership to each component, together with the conventional density and orientation variables. The convergence towards the non-fractional, zero/one membership during optimization is achieved by a hypercube-to-simplex projection of the membership variables with a penalization scheme. Through the integration with a continuous material orientation design method, the proposed method is capable of generating multi-component composite structures with tailored material orientations for each component, without a prescribed set of alternative discrete angles. The new method extends the existing continuous material orientation design methods by introducing the concept of components (substructures) with the component-level orientation control, which is suitable for economical production with the conventional high-volume composite manufacturing processes.

Challenging problems of computational mechanics may often be characterized by large deformations that are common in manufacturing processes such as forging. The finite element method faces difficulties in simulating large deformations, due to severe mesh distortion . A solution to overcome these difficulties is to use meshless methods like Smoothed Particle Hydrodynamics (SPH). This paper presents a thermomechanical SPH in total Lagrangian formulation to simulate efficiently large deformations thermomechanical problems. The continuum is modeled as a Hamiltonian system of particles (energy-based framework) when dissipative effects are considered where the constitutive equation is represented via an internal energy term. A comparison with an Eulerian SPH formulation and FEM is presented to assess the accuracy of the total Lagrangian formulation through examples of high velocity Taylor impact test and hot forging test.

In this work, a local moving extended phase field method (LMXPFM) is proposed for the failure analysis of brittle materials. In this approach, nonlinear phase field crack evolution equations are solved in an adaptively refined small region (which is prone to crack nucleation or propagation). In the proposed LMXPFM, phase field method (PFM), extended finite element method (XFEM) and multiscale finite element method (MsFEM) are combined together such that the MsFEM acts as a medium to couple coarse mesh with fine mesh, and the XFEM is used to represent a discontinuity (such as crack) outside the phase field region. Thus, the LMXPFM not only provides an adaptively refined mesh in a small region near the smeared discontinuity but also fulfills the prerequisite of refined mesh for solving phase field crack evolution equations. This local approach provides a huge saving in the computational efforts as compared to PFM applied in the entire domain. The efficiency and accuracy of the LMXPFM are verified by solving and comparing the results with standard PFM and literature solutions.

Layout design of multi-component systems has received ever-increasing attention over recent years. However, most of the current literature on the layout design of multi-component systems conducted are all carried out in an implicit way. In this paper, an explicit optimization model based on the moving morphable bars method is proposed. To this aim, topology description functions are used to describe the geometrical shapes of embedded components, while moving bars act as supporting structure that connects these embedded components. Different from traditional optimization methods, the geometric parameters used to describe the size, shape, location and orientation of the moving bars and embedded components are considered as design variables in this work. To avoid remeshing the grids and improve the efficiency of computation, the moving bars and embedding components are mapped into two density fields on a fixed grid using a smoothed Heaviside function . A discrete material interpolation scheme developed for orientation optimization problem is extended for the first time to the material parameterization of multi-component systems. Moreover, a single explicit constraint based on the volume and perimeter of embedded components is proposed for avoiding overlaps between the embedded components, and between each component and the design domain boundary. Several numerical examples are performed to show the effectiveness and flexibility of the presented optimization model in handling layout optimization of structures with multi-phase embedded components.

In this study, a general frame of the forward and inverse structural uncertainty propagations (UPs) based on the dimension reduction (DR) method and the derivative lambda probability density function ( λ -PDF) is proposed to handle arbitrary probability distribution. For the forward UP, a DR method is applied to decompose a multivariable system into multiple univariate subsystems and a derivative λ -PDF is adopted to transform the arbitrary probability distribution of each subsystem. Then, the statistical moments and a fitting region are mathematically derived to analyze the fitting ability of the derivative λ -PDF. According to whether the kurtosis–skewness point lies in or out the fitting region, two different strategies combining the Gauss–Gegenbauer quadrature are proposed to implement the forward propagation. Compared with the conventional methods, the proposed method has advantages in higher accuracy, stability and efficiency. For the inverse propagation, because the unknown variable may be arbitrary distribution, the general frame based on the derivative λ -PDF and the Gauss–Gegenbauer quadrature are utilized to convert the uncertainty propagation into multiple deterministic calculations. Based on this, optimization method is adopted to accurately obtain the statistical moments and PDFs of the unknown stochastic variables. Five examples are provided to verify the accuracy and efficiency of the proposed general frame for the forward and inverse UPs.

A multiscale modeling and characterization method for coupled damage-healing-plasticity occurring in granular material is proposed. The characterization is performed on the basis of multiscale modeling of granular material in the frame of concurrent second-order computational homogenization method, in which granular material is modeled as gradient-enhanced Cosserat continuum at the macroscale. The damage-healing-plasticity is characterized in terms of meso-structural evolution of discrete particle assembly within representative volume elements (RVEs) assigned to selected local material points in macroscopic continuum, with no need to specify macroscopic phenomenological constitutive models, failure criteria along with evolution laws, and associated macroscopic material parameters. The proposed modeling and characterization method for coupled damage-healing-plasticity in granular material is comprised of the following three constituents. The incremental non-linear constitutive relation for the discrete particle assembly of RVE is first established. Then the meso-mechanically informed incremental non-linear constitutive relation of macroscopic gradient-enhanced Cosserat continuum is derived from volume averages of the RVE-scale solutions. Finally the thermodynamic framework is set up to define meso-mechanically informed anisotropic damage and healing factors, anisotropic net damage factors combining both damage and healing effects, and plastic strains. Densities of damage, plastic and total dissipative energies as well as non-dissipative healing energy, as scalar internal state variables, are provided to compare the effects of damage, healing and plasticity on material failure and structural collapse. The numerical example of strain localization and softening problem is performed to demonstrate the performance and applicability of the proposed multiscale modeling and characterization method of coupled damage-healing-plasticity for granular materials.

It is often highly desirable to simultaneously optimize the layout of embedded functional components and the topology of the host structure supporting these components to achieve the best overall performance while still ensuring the structural integrity. We propose a topology optimization framework to account for connecting interface behaviors between the components and the host structure. Here we treat the connecting interfaces with the cohesive zone model to reflect the adhesively bonded interface behaviors. A conforming mesh in conjunction with interface elements is employed to discretize the evolving structure while accounting for the strong discontinuity of displacement field across the material interfaces . To give a clear representation of structural boundaries and the connecting interfaces, we also suggest a multi-material interpolation model in the level set framework, which can conveniently define the connecting interface locations and describe multi-material distribution without redundant phase in the design domain. The objective function is defined as the sum of the strain energy and the work done by the traction on the connecting interface, and the evolution velocities of the level set and the embedded components are treated as design variables. These design variables are updated with the MMA optimizer on the basis of adjoint-variable sensitivity analysis. This optimization formulation allows multiple constraints and mixed design variables ( i.e level set design variables and components’ design variables) to be easily handled in level set based optimization. The design is advanced by the Hamilton–Jacobi equation with the velocity design variables as input. Numerical examples demonstrate the validity and applicability of the proposed method.

With the development of reliability technique, the safety assessment for the problem with epistemic uncertainty has attracted widespread attention. Evidence theory is a useful tool to deal with such uncertainty, and this paper aims to develop an efficient approach for the evidence theory-based reliability analysis and optimization design. By using mutually exclusive intervals to quantify the focal elements of evidence variable, a confidence range bounded by belief measure and plausibility measure is derived for system reliability assessment , by which the relatively conservative and radical optimization models can be respectively established. To decrease the huge computational burden in repetitive limit state function evaluations under the time-consuming implicit computational model, an explicit surrogate model is constructed by the Legendre polynomial chaos expansion in the support box. A Clenshaw–Curtis point-based collocation method with Smolyak algorithm is then developed to predict the unknown coefficients in surrogate model, where the collocation level can be flexibly selected according to the accuracy requirement. Compared with the traditional deterministic optimization model under nominal value assumption, the results in two numerical examples verify the effectiveness of proposed method in mathematical theory and engineering application.

In this study, a new axisymmetric ordinary state-based peridynamic (PD) model for axisymmetric problems of linear elastic solids is presented. A fracture criterion based on the PD bond energy density is proposed. Adaptive dynamic relaxation (ADR) method is adopted to obtain equilibrium solutions, and a viable fictitious density of the model is derived and proven to be valid for implementation in the ADR method. Performance and validity of the proposed axisymmetric PD model are demonstrated by three kinds of numerical problems, i.e., compression tests, pull-out deformation, and indentation fracture. In the compression tests with focus on constant-strain deformations, the displacements predicted by the present model are compared with classical analytical solutions, and the results show good agreements. Both m- convergence and δ -convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases. The recovery of the Poisson’s ratios in the model is tested in detail as well. The capability of capturing general non-uniform axisymmetric deformation by the proposed model is verified in the pull-out analysis. The equilibrium displacement fields predicted by the present model agree very well with those by the finite element method. The peridynamic evaluation of strains and stresses also show good match with the finite element ones. The proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data. The model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state-based peridynamics.

This paper presents a simple and effective strain smoothing method (the strain-smoothed element method) for 3-node triangular and 4-node tetrahedral solid finite elements. While piecewise constant strain fields are constructed through smoothing domains in previous strain smoothing methods, in the proposed method, linear strain fields are constructed within finite elements using constant strains of neighboring finite elements. When the new strain smoothing method is adopted, the triangular and tetrahedral solid finite elements pass all the basic tests; patch, isotropy and zero energy mode tests, and show improved predictive capability. The formulation of the proposed method is presented in detail. Through various numerical examples, we demonstrate the accuracy improvement achieved.

In this work we develop a multi-scale model to characterise the large scale constitutive behaviour of a material featuring a small scale fibrous architecture. The Method of Multi-scale Virtual Power (MMVP) is employed to construct the model. At the macro-scale, a classical continuum mechanics problem is formulated in the finite strain regime. At the micro-scale, a network of fibres, modelled as one-dimensional continua, composes the representative volume element (RVE). The MMVP provides a full characterisation of the equilibrium problem at the RVE, with consistent boundary conditions, as well as the homogenisation formula which defines the first Piola–Kirchhoff stress tensor. Particular attention is given to the fact that the macro-scale continuum could be considered incompressible. Numerical experiments are presented and model consistency is verified against well-known phenomenological constitutive equations. Scenarios departing from the hypotheses of such phenomenological material models are discussed. Fil: Rocha, Felipe Figueredo. Laboratório Nacional de Computação Científica; Brasil. Instituto Nacional de Ciência e Tecnologia em Medicina Assistida por Computação Científica; Brasil Fil: Blanco, Pablo Javier. Laboratório Nacional de Computação Científica; Brasil. Instituto Nacional de Ciência e Tecnologia em Medicina Assistida por Computação Científica; Brasil Fil: Sánchez, Pablo Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones en Métodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones en Métodos Computacionales; Argentina. Universidad Tecnológica Nacional; Argentina Fil: Feijóo, Raúl Antonino. Laboratório Nacional de Computação Científica; Brasil. Instituto Nacional de Ciência e Tecnologia em Medicina Assistida por Computação Científica; Brasil

We propose an isogeometric approach for flexoelectricity in soft dielectric materials at finite deformations accounting for Maxwell stresses on the surface between two different media. In contrast to piezoelectricity where there is a linear dependence between mechanical strain and electric polarization , in flexoelectricity the polarization is related to strain gradients which requires a C 1 continuous finite element framework. In electro-mechanical materials, the Maxwell stress emerges as a consequence of the coupling effect between electrostatics and mechanics. If a solid body is embedded in a surrounding medium such as air or vacuum, the Maxwell stress acting on the surfaces governs the interaction between electric fields and deformable media. This is quite difficult to handle due to the surface discontinuity and the traction electrical forces still have to satisfy some certain continuity conditions , hence an appropriate numerical framework is required for the treatment. Here, we employed Non-Uniform Rational B-spline (NURBS) functions with knot insertion technique in order to introduce discontinuities across material interfaces while still maintaining C 1 continuity in the domain to evaluate the coupling effect of strain gradient and electric polarization in the regime of finite deformation. The accuracy and robustness of our IGA approach for flexoelectric soft materials are demonstrated in some benchmark numerical examples.

The well-known phase-field approach applied to fracturing solids has recently been embedded in the Theory of Porous Media for the description of hydraulic fracturing scenarios. This method has been found very convenient not only for the description of solid deformation and fracture but also for the transition of Darcy-type flow in the saturated porous-media domain towards Navier–Stokes-type flow in fractured zones. However, as a result of the monotonic evolution of the phase-field variable, the phase-field approach to fracture does not usually allow for the description of pre-existing closed fractures or of fractures closing after generation, where either only Darcy-type flow occurs or where Navier–Stokes-type flow turns back to Darcy-type flow. To tackle this issue, the present study concerns the introduction of a crack-opening indicator as an additional variable governed by the current deformation. By use of this procedure, not only opening but also closing fractures as well as pre-fractured domains can easily be included into the numerical simulation of fracking scenarios in saturated porous media. Proceeding from the finite-element analysis, the numerical results are found consistent with experimental observations presented in the literature.

This work addresses an efficient low order virtual element method (VEM) for the phase-field modeling of isotropic brittle fracture. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with constitutive terms rooted in fracture mechanics, see Miehe et al. (2010) and Miehe et al. (2015) . In the presented contribution, we propose a rigorous variational-based framework for the phase-field modeling of brittle fracture in elastic solids undergoing small strains. The key goal here, is the extension towards the recently developed virtual element formulation due to the flexible choice of nodes number in an element which can be changed easily during the simulation process, as outlined in Wriggers et al. (2016) and Wriggers et al. (2017). To this end, the potential density is formulated in terms of suitable polynomial functions, instead of computing the unknown shape functions for complicated element geometries, e.g. arbitrary convex or concave polygonal elements. An important aspect of this work is the introduction of an incremental minimization principle, with a novel construction of the stabilization density for the coupled multi-field problem. On the computational side, a robust and efficient monolithic scheme is employed using the software tool AceFEM program in the numerical implementation to compute the unknowns (displacement and crack phase-field), see Korelc and Wriggers (2016). The performance of the formulation is underlined by means of representative examples.

International audience; One of the challenges of fatigue simulation using continuum damage mechanics framework over the years has been reduction of numerical cost while maintaining acceptable accuracy. The extremely high numerical expense is due to the temporal part of the quantities of interest which must reflect the state of a structure that is subjected to exorbitant number of load cycles. A novel attempt here is to present a non-incremental LATIN-PGD framework incorporating temporal model order reduction. LATIN-PGD method is based on separation of spatial and temporal parts of the mechanical variables, thereby allowing for separate treatment of the temporal problem. The internal variables, especially damage, although extraneous to the variable separation, must also be treated in a tactical way to reduce numerical expense. A temporal multi-scale approach is proposed that is based on the idea that the quantities of interest show a slow evolution along the cycles and a rapid evolution within the cycles. This assumption boils down to a finite element like discretisation of the temporal domain using a set of "nodal cycles" defined on the slow time scale. Within them, the quantities of interest must satisfy the global admissibility conditions and constitutive relations with respect to the fast time scale. Thereafter, information of the "nodal cycles" can be interpolated to simulate the behaviour on the whole temporal domain. This numerical strategy is tested on different academic examples and leads to an extreme reduction in numerical expense.

In conventional aeroelastic analysis and optimisation methods for wing design, different geometries are used for the different steps in the process. Generally, a parametrised model is used to describe the shape of the geometry for the optimisation process. Subsequently, this model is converted into a structural and aerodynamic model for analysis purposes. These steps increase the computational effort and introduce geometrical errors. In this work, a geometrically consistent static aeroelastic analysis framework is presented. By using isogeometric analysis, the exact geometry is used in both the structural and aerodynamic models, preventing any additional computational effort for meshing and geometry retrieval steps and avoiding the introduction of geometrical errors due to the discretisation of the geometry. The separate components of the framework are described and verified, and the complete framework is demonstrated through the analysis of the realistic wing model.

Recently several non-standard phase-field models different from the standard one for brittle fracture (Bourdin et al. 2000, 2008; Miehe et al. 2010a) have been proposed for the modeling of damage and fracture in solids. On the one hand, linear elastic behavior before the onset of damage and even cohesive cracking induced quasi-brittle failure can be considered by such non-standard phase-field models. On the other hand, they present great challenges to the numerical implementation since the damage boundedness is no longer automatically fulfilled. In this work, the numerical implementation of the unified phase-field damage model (Wu 2017, 2018) is addressed, though it also applies to the standard and other non-standard ones. In particular, an iterative alternate minimization (AM) algorithm enhanced with path-following strategies is presented. The phase-field or damage sub-problem is solved by the bound-constrained optimization solver in which the boundedness and irreversibility conditions of the crack phase-field are exactly enforced. Moreover, material softening induced snap-backs typically for localized failure in solids can also be effectively dealt with. The equilibrium paths in terms of the fracture surface and of the indirect displacement (e.g., the crack mouth opening or sliding displacement) are discussed in the context of the AM algorithm. The AM algorithm with the indirect displacement control is advocated, since both prescribed external forces and displacements can be naturally dealt with, which is more versatile than the one with the fracture surface control applicable only for prescribed displacements. Representative examples of benchmark tests show that the AM algorithm enhanced with the indirect displacement control is extremely robust even for large increment sizes. The insensitivity of the unified phase-field damage model to the incorporated length scale and mesh size is also confirmed.

Symmetry and repetitive patterns are very common in both natural and artificial structures. Design with symmetry and pattern repetition (SPR) can be very useful in practical engineering. In this paper, we discuss a new optimal structural design method considering SPR constraints. The basic idea consists in combining a parameterized level-set method together with radial basis function (RBF) interpolation strategy, which allows for decoupling RBF knots from the finite element (FE) analysis mesh. The merits of the decoupled manner are that RBF knots can be mapped independently on regular or irregular design domain, and SPR constraints can be imposed by simply building a bijective relationship between RBF knots regardless of analysis mesh. Furthermore, due to parameterization of the level-set function, there are fewer numerical manipulations that need to be implemented than the conventional level-set method during the optimization process. Several numerical examples are presented to show feasibility and effectiveness of the proposed method.

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

A framework is presented for multi-material compliance minimization in the context of continuum based topology optimization . We adopt the common approach of finding an optimal shape by solving a series of explicit convex (linear) approximations to the volume constrained compliance minimization problem . The dual objective associated with the linearized subproblems is a separable function of the Lagrange multipliers and thus, the update of each design variable is dependent only on the Lagrange multiplier of its associated volume constraint. By tailoring the ZPR design variable update scheme to the continuum setting, each volume constraint is updated independently. This formulation leads to a setting in which sufficiently general volume/mass constraints can be specified, i.e., each volume/mass constraint can control either all or a subset of the candidate materials and can control either the entire domain (global constraints) or a sub-region of the domain (local constraints). Material interpolation schemes are investigated and coupled with the presented approach. The key ideas presented herein are demonstrated through representative examples in 2D and 3D.

In this paper, we present a method that combines the Boundary Element Method (BEM) with IsoGeometric Analysis (IGA) for numerically solving the system of Boundary Integral Equations (BIE) arising in the context of a 2-D steady-state heat conduction problem across a periodic interface separating two conducting and conforming media. Our approach leads to a fast solver with high convergence rate when compared with low-order BEM. Additionally, an optimization framework comprising a parametric model for the interface’s shape, our IGA-BEM solver, and evolutionary and gradient-based optimization algorithms is developed and tested. The optimization examples demonstrate the efficiency of the framework in generating optimum interfaces for maximizing heat transfer under various geometric constraints.

A superconvergent meshfree collocation method with smoothed nodal gradients is presented. In this method, the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function, while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients. It is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function, which facilitates a trivial numerical implementation. Subsequently, an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points. Based upon a local truncation error analysis, it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions. A key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation. Another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation, which is non-feasible in the conventional collocation formulation. The effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems. Numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method.

The most common discretization method for peridynamic models used in engineering problems is the node-based meshfree approach. This method discretizes peridynamic domains by a set of nodes, each associated with a nodal cell with a characteristic volume, leading to a particle-based description of continuum systems. The behavior of each particle is then considered representative of its cell. This limits the convergence rate to the first order. In this paper, we introduce a reproducing kernel (RK) approximation to the field variables in the peridynamic equations to increase the order of convergence of peridynamic numerical solutions. The numerical results demonstrate improved convergence rates in static peridynamic problems using the proposed method.

In order to optimally design materials, it is crucial to understand the structure–property relations in the material by analyzing the effect of microstructure parameters on the macroscopic properties . In computational homogenization , the microstructure is thus explicitly modeled inside the macrostructure, leading to a coupled two-scale formulation. Unfortunately, the high computational costs of such multiscale simulations often render the solution of design, optimization, or inverse problems infeasible. To address this issue, we propose in this work a non-intrusive reduced basis method to construct inexpensive surrogates for parametrized microscale problems; the method is specifically well-suited for multiscale simulations since the coupled simulation is decoupled into two independent problems: (1) solving the microscopic problem for different (loading or material) parameters and learning a surrogate model from the data; and (2) solving the macroscopic problem with the learned material model. The proposed method has three key features. First, the microscopic stress field can be fully recovered, which is useful for instance for revealing local stress concentrations inside the microstructure. Second, the method is able to accurately predict the stress field for a wide range of material parameters; furthermore, the derivatives of the effective stress with respect to the material parameters are available and can be readily utilized in solving optimization problems . Finally, it is more data efficient, i.e. requiring less training data, as compared to directly performing a regression on the effective stress. To construct the surrogate model, first, a proper orthogonal decomposition is performed on precomputed microscopic stress field snapshots to find a reduced basis for the stress. Second, a regression is employed to infer the coefficients of the reduced basis approximation for any arbitrary parameter value, thus enabling a rapid online evaluation of the microscopic stress. Equipped with the stress field, the effective stress and its partial derivatives can then be derived analytically. For the microstructures in the two test problems considered, the mean approximation error of the effective stress is as low as 0.1% despite using a relatively small training dataset . Embedded into the macroscopic problem, the reduced order model leads to an online computational speed up of approximately three orders of magnitude while maintaining a high accuracy as compared to the FE 2 solver.

Recent developments in numerical simulations in the aerodynamics field are focused on reducing the computational cost of the solvers, targeting their use at the initial design steps. Under the right assumptions, potential solvers can be exploited to accomplish a valid solution of the flow field for streamlined bodies in subsonic and close to transonic flows. A fully embedded approach to solve the full-potential equation is presented, where both the geometry and the wake are defined implicitly with a level set function. Embedded methods allow simplifying the mesh generation process, as only a background mesh is required to perform the analysis. The presented method gives an automatic and fast option to solve subsonic flows, which is especially desirable in optimization workflows. This work has been supported by the European Commission through the Horizon 2020 Research and Innovation program under contract 800898 (ExaQUte project). The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness , through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2018-000797-S).

Phase-field models for fracture have gained exceptional popularity in the last couple of decades and have been extended into areas well beyond brittle quasi-static fracture propagation to ductile, dynamic, or hydraulic fracturing. Despite the significant theoretical advancements in these more complex physical settings, little attention has been paid to the quantification of crack opening displacement to date as most applications do not explicitly require the crack opening displacement for the morphological evolution of cracks. However, one of the exemptions would be hydraulic fracturing where the crack propagation is driven by the fluid pressure which strongly depends on the crack opening displacement. In this study, we look into two known approaches, a line integral and a level-set method, for crack opening computation mainly from an implementation point of view. Firstly, we derive an approximation of a discontinuous function field in the variational phase-field setting which is then applied to obtain the crack opening (displacement jump) and verify the approximation against a closed solution. We then propose a “certain distance from the crack” required for the level-set function using a one-dimensional analysis. Finally, we compare these approaches under several different conditions such as crack alignment to the mesh or under loading (asymmetric displacement field) and investigate the convergence with respect to the mesh size.

Selective Laser Melting (SLM) is an Additive Manufacturing (AM) process where a powder bed is locally melted. Layer by layer, complex three dimensional geometries including overhangs can be produced. Non-melted powder thereby acts as support structure. The process is held under an inert gas atmosphere to prevent oxidation. The principal machine parameters in SLM processes are the laser power, the scan rate and the laser spot radius. The powder bed is characterized by the material, the packing density and the particle size distribution. These factors define the structure of SLM finished parts. Up to date, the material and process development of SLM mainly relies on experimental studies that are time intensive and costly. Simulation tools offer the potential to gain a deeper understanding of the process–structure–property interaction. This can help to find optimal process parameters and to individualize AM manufactured parts. A continuum framework for the finite deformation phase change problem is developed. For its numerical solution the stabilized Optimal Transportation Meshfree Method (OTM) is employed. The advantage of meshfree over conventional mesh based techniques is that the treatment of particle fusion is intrinsic to the formulation. This is important to resolve the complex moving boundaries between liquid melt flow and solid metal. In a numerical example consisting of two metal powder particles, the influence of laser heating and cooling conditions on melting and consolidation is analyzed. A detailed parameter study is presented. The insight gained from the simulations may help to narrow the parameter window for further investigations.

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

An adaptive multiscale approach for identifying multiple flaws in structures is proposed in this paper. The approach includes a two-step process in which a coarse-scale search initially identifies the candidate subdomains and then a fine-scale search of the narrow subdomains captures the true flaws. The extended finite element method (XFEM) is employed to solve the forward problem because it does not require re-meshing in each iteration. A discrete artificial fish swarm (DAFS) algorithm with adaptive vision is adopted to accelerate the convergence of the objective function. The probable subdomains are output via K-means clustering with an initial estimated number of flaws Then, the artificial bee colony (ABC) algorithm is employed to capture the true flaws and remove the false flaws. The DAFS algorithm is a single-population algorithm, while the ABC algorithm is a multi-population algorithm. Three numerical examples are studied to evaluate the accuracy and performance of the proposed approach. Circular and elliptical flaws and the effects of measurement noise are considered to test the robustness. The results show that this approach can effectively quantify and identify multiple internal flaws without any prior knowledge of their quantity. Furthermore, the strategy is more efficient than previously proposed approaches.

Non-probabilistic reliability-based design optimization (NRBDO) is an essential tool to deal with an optimization model with uncertain-but-bounded variables. A major disadvantage of NRBDO is that it conventionally suffers from excessively high computational costs. In the present study, a new target performance approach (TPA) is proposed to eliminate high computational cost incurred by complicated equality constraints in non-probabilistic reliability analysis in which the target performance constraint is applied to substitute for the non-probabilistic reliability index constraint. To further enhance the optimization efficiency, a new NRBDO algorithm is developed based on the proposed TPA in which the target performance iterative method (TPIM) is established to search for the target concern point while a convex approximate method is used to further improve the convergence. A mathematical example, three numerical examples, and two complex engineering examples, i.e., an axially compressed stiffened shell in a launch vehicle and a tower crane in civil engineering, are utilized to demonstrate the effectiveness of the proposed method in comparison with that of other existing methods. The results indicate that the proposed method is promising in terms of complicated engineering problems in the absence of prior knowledge of uncertain parameters.

We present an effective adaptive analysis procedure in terms of isogeometric analysis (IGA) based on locally refined (LR) B-splines for two-dimensional elasticity problems. One major advantage of the adopted LR B-splines over B-splines and non-uniform rational B-splines (NURBS) is the local refinement, which makes it highly suitable for adaptive analysis. The Zienkiewicz–Zhu estimation is used on error estimator based local refinement. The local refinement is implemented by the structured mesh refinement strategy according to the posteriori error estimator. The proposed methodology is further extended to deal with multiple patches, which are applied to represent complicated shapes of structures, and the Nitsche’s method is employed for coupling patches among multiple patches in the present adaptive IGA. The algorithm for adaptive IGA on multiple patches is also derived. We provide a series of numerical examples for benchmarks and complicated configurations to demonstrate the flexibility and accuracy of the model as well as the significant enhancement in adaptive IGA based on LR B-splines. A careful comparison of the numerical results suggests that the adaptive local refinement strategy offers faster convergence rate than that of uniform refinement strategy.

The traditional analysis and design of the phononic crystals (PhCs) are based on the deterministic physical models. However, uncertainties, unavoidably existing in the PhCs, seriously affect their physical properties. To quantify the influence of uncertainties on the band structures of the PhCs, three uncertain models, including an interval model, a random model and a hybrid uncertain model, are considered. In order to overcome the considerably computational cost of the evaluation of the band structures of the PhCs, a new high-order sparse Gegenbauer polynomial surrogate model (HOSGPSM) is proposed. In the HOSGPSM, a new spare sequential sampling scheme based on the maximin principle is employed in the sampling process, greatly reducing the potential samples. And then, the least square method is introduced to calculate the unknown coefficients of the HOSGPSM. According to the orthogonal property of the Gegenbauer polynomial, the expectation and variance of the HOSGPSM with respect to random variables can be directly yielded. The bounds of the HOSGPSM with respect to the interval model and the hybrid uncertain model can be obtained by the Monte Carlo method . The numerical results of three typical PhCs show that the HOSGPSM can accurately and efficiently evaluate the band structures of the PhCs under three uncertain models, providing a unified platform for the numerical analysis and the optimal design of various uncertain PhCs.

This paper introduces a new optimization framework to multiscale structures with unknown-but-bounded (UBB) uncertainties, where the optimal design is applied at two scales: the macroscale, i.e., the structural level, where the non-probabilistic reliability-based topology optimization (NRBTO) is conducted, and the microscale, i.e., the material level, where the non-probabilistic reliability-based design optimization (NRBDO) for truss-like cell dimensions is achieved. By combination of the homogenization method and interval mathematics, rules of macro–micro stiffness equivalence as well as interval bounds of macro displacement responses under specific micro structural settings can be first confirmed. For safety reasons, novel reliability definitions are respectively given for the double-level design, in which the offset factor with good differential properties is applied for the NRBTO and the matching factor derived from the improved rule of interval sorting is utilized for the NRBDO. To circumvent problems of large-scale variable updating in two-level optimization procedures, the adjoint-based vector in macro layout design as well as the particle swarm optimization (PSO) algorithm in micro parametric design are further discussed. The usage, effectiveness, and the superiority of the developed methodology are eventually demonstrated by two 3D engineering applications.

The Timoshenko beam bending problem is formulated in the context of strain gradient elasticity for both static and dynamic analysis. Two non-standard variational formulations in the Sobolev space framework are presented in order to avoid the numerical shear locking effect pronounced in the strain gradient context. Both formulations are shown to be reducible to their locking-free counterparts of classical elasticity. Conforming Galerkin discretizations for numerical results are obtained by an isogeometric C p − 1 -continuous approach with B-spline basis functions of order p ≥ 2 . Convergence analyses cover both statics and free vibrations as well as both strain gradient and classical elasticity. Parameter studies for the thickness and gradient parameters, including micro-inertia terms, demonstrate the capability of the beam model in capturing size effects. Finally, a model comparison between the gradient-elastic Timoshenko and Euler–Bernoulli beam models justifies the relevance of the former, confirmed by experimental results on nano-beams from literature.

This paper focuses on the problem of combined state/parameter estimation in dynamical systems with many degrees of freedom, for instance finite element models, using measured data from the system and nonlinear Bayesian filtering techniques for the estimation. A highly efficient nonlinear Kalman filtering technique is developed, based on a combination of linear or extended Kalman filtering for state estimation and unscented filtering for parameter learning. Such a combination is implemented by applying the principle of marginalization to the unscented transform. This method leads to a very efficient state/parameter filtering algorithm by taking advantage of the fact that Jacobians of the system equations with respect to the dynamic states, required in extended Kalman filtering, can be easily related to outputs of the finite element analysis required for forward propagation. For linear dynamical systems this approach yields an algorithm with identical accuracy as the UKF but reduced computational time, as demonstrated on several medium-size examples. For nonlinear systems, the resulting algorithm is also superior to the UKF in terms of computational time, due to the fact that Jacobians are functions of the tangent stiffness matrix of the system, whose computation is required for propagation in finite element analysis. It is also shown that, even though this algorithm relies on linearization for propagation of moments of the dynamic states, this reduction in accuracy compared to a generic UKF does not affect learning of the parameters for the systems considered herein.

Non-ordinary state-based peridynamics suffers from zero-energy modes due to nodal integration, causing instabilities of the displacement, stress and strain fields. A stabilized non-ordinary state-based peridynamic model is derived according to the linearized bond-based peridynamic theory. The incorporation of supplemented force state excludes a zero-energy modes control coefficient and eliminates the need for complicated parameter adjustment. Finally, the parameter of strain energy ratio is defined to study the degree of influence of the zero-energy modes on the computation process. Four numerical examples are analyzed to demonstrate the effectiveness of the present model in controlling the zero-energy modes in non-ordinary state-based peridynamics.

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points . Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO.

We propose a solution strategy for a multimaterial minimum compliance topology optimization problem, which consists in finding the optimal allocation of a finite number of candidate (possibly anisotropic) materials inside a reference domain, with the aim of maximizing the stiffness of the body. As a relevant and novel application we consider the optimization of self-assembled structures obtained by means of diblock copolymers. Such polymers are a class of self-assembling materials which spontaneously synthesize periodic microstructures at the nanoscale , whose anisotropic features can be exploited to build structures with optimal elastic response , resembling biological tissues exhibiting microstructures, such as bones and wood. For this purpose we present a new generalization of the classical Optimality Criteria algorithm to encompass a wider class of problems, where multiple candidate materials are considered, the orientation of the anisotropic materials is optimized, and the elastic properties of the materials are assumed to depend on a scalar parameter , which is optimized simultaneously to the material allocation and orientation. Well-posedness of the optimization problem and well-definition of the presented algorithm are narrowly treated and proved. The capabilities of the proposed method are assessed through several numerical tests.

Numerical formulations of the Koiter theory allow the efficient prediction, through a reduced model, of the behavior of shell structures when failure is dominated by buckling. In this work, we propose an isogeometric version of the method based on a solid-shell model. A NURBS interpolation is employed on the middle surface of the shell to accurately describe the geometry and the high continuity typical of the displacement field in buckling problems and to directly link the CAD model to the structural one. A linear interpolation is then adopted through the thickness together with a modified generalized constitutive matrix, which allows us to easily eliminate thickness locking and model multi-layered composites. Reduced integration schemes, which take into account the continuity of the shape functions, are used to avoid interpolation locking and make the integration faster. A Mixed Integration Point strategy makes it possible to transform the displacement model into a mixed (stress–displacement) one, required by the Koiter method to obtain accurate predictions, without introducing stress interpolation functions. The result is an efficient numerical tool for buckling and initial post-buckling analysis of composite shells, characterized by a low number of DOFs and integration points and by a simple and quick construction of the reduced model.

This study presents the weak form of peridynamic (PD) governing equations which permit the direct imposition of nonlocal essential and natural boundary conditions. It also presents a variational approach to derive the PD form of first-and secondorder derivatives of a field variable at a point which is not symmetrically located in its domain of interaction. This capability enables the nonlocal PD representation of the internal force vector and the stress components without any calibration procedure. Furthermore, it removes the concern of truncated domain of interaction for a point near the surface. Thus, the solution is free of nonlocal boundary forces and surface effects. The numerical solution of the resulting equations can be achieved by considering an unstructured nonuniform discretization. The implicit solution to the discrete form of the equations is achieved by employing BiConjugate Gradient Stabilized (BICGSTAB) method which is an iterative technique for solving sparse non-symmetric linear systems. The explicit analysis is performed by constructing a global diagonal mass matrix, and using a hybrid implicit/explicit time integration scheme. The accuracy of this approach is demonstrated by considering an elastic isotropic plate with or without a cutout subjected to a combination of different types of boundary conditions under plane stress conditions. (C) 2018 Elsevier B.V. All rights reserved.

A novel procedure for analyzing fracture processes in quasi-brittle materials and consisting in combining the discrete crack approach by means of interface elements with the Virtual Element Method is proposed and developed in this work. In particular, the proposed procedure is used in the simulation of non-linear mechanical response and cracking of cement-based composites at the mesoscopic level of observation. Thereby three components are recognized: mortar, coarse aggregates and mortar–aggregate interfaces. In this regard VEM constitutes a powerful and efficient tool to represent the complex geometries of the inclusions in composite materials, such as coarse aggregates in concrete. Actually, patches with any number of edges (not necessarily convex), hanging nodes, flat angles, collapsing nodes, etc., can be easily handled in the VEM framework while retaining the same approximation properties of FEM. On the other hand, classical zero-thickness interface elements (IEs) are employed for modeling stress-crack opening processes. A series of numerical results, not only at the mesoscopic but also at the macroscopic level of observation, are presented to demonstrate the soundness and capabilities of the proposed approach based on combinations of VEM and IEs. Fil: Benedetto, Matías Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina Fil: Caggiano, Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina. Technische Universität Darmstadt; Alemania Fil: Etse, Jose Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina. Universidad Nacional de Tucumán; Argentina

A novel constitutive framework suitable for material interfaces undergoing large deformations in a geometrically exact setting was developed in Ottosen et al. (2016). In contrast to previous works, it permits the description of arbitrary material anisotropies by fulfilling all fundamental balance laws in physics as well as the principle of material objectivity. This paper deals with an efficient finite element implementation of the aforementioned framework in terms of the natural basis vectors. To be more precise, a different, more compact and more direct derivation of this framework is outlined first. It relies on the variational structure of the underlying problem. Subsequently, the aforementioned finite element approximation is elaborated which is finally embedded into a computational homogenization scheme. This scheme allows the analysis of the influence of the novel interface model on the resulting macroscopic (effective) material response. It is shown by numerical examples that the interaction of bulk energies and interface energies leads, in a very natural manner, to a complex size effect. It includes the frequently observed “the smaller the stiffer” relation, but also the less often observed “the smaller the softer” relation. However, since the overall response is usually a superposition of such relations, the effective properties cannot generally be characterized by one of the aforementioned limiting relations.

The diameter of spot welds used in the automotive and aerospace industry is orders of magnitude smaller than the dimensions of the structural components. Automotive bodies typically have between three and five thousand spot welds which makes 3-D Direct Finite Element Analysis (DFEA) of this class problem not practical. To circumvent these limitations of the FEM, spot welds are idealized in industrial problems as elastic or rigid beams connecting nodes of the metal sheet meshes. However, parameters used in these spot weld models are often problem dependent. This paper presents a parallel Generalized Finite Element Method (GFEM) for the simulation of spot welds in large structures. The proposed GFEM can adopt structural-scale meshes that ignore the spot welds and thus can be generated much quicker than in a DFEA. The element size in the GFEM mesh is of the same order as those used in the industry for the class of problems considered here. Verification problems show that the proposed method can provide an accuracy comparable to a 3-D DFEA. Numerical experiments on a hat-stiffened panel with 168 spot welds show that the proposed GFEM scales much better than a DFEA on shared memory machines, in particular when nonlinear material behavior near the spot welds is considered.