The FFT-based homogenization method of Moulinec & Suquet has been established as a fast, accurate and robust tool for periodic homogenization in solid mechanics. In a finite element context, Pahr & Zysset have introduced non-periodic boundary conditions (PMUBC) for homogenization problems. We show how to implement PMUBC efficiently in an FFT-based code using discrete sine and cosine transforms. Compared

This paper presents an adaptive multi-surrogate constrained optimization method (AMSCOM) that can automatically determine the appropriate metamodel for each black-box function in the constrained optimization problem (COP) and concurrently find the optimum. In AMSCOM, each black-box function is approximated initially by several different types of candidate surrogates. Then, as optimization progresses

In this work, a greedy reduced basis algorithm is proposed for the solution of structural acoustic systems with parameter and implicit frequency dependence. The underlying equations of linear time-harmonic elastodynamics and acoustics are discretized using the finite element and boundary element method, respectively. The solution within the parameter domain is determined by a linear combination of

Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order

This paper presents an accurate porosity-velocity-concentration approach, in which porosity, pore-fluid velocity and the concentration of dissolvable substances in the pore fluid are selected as four primary unknown variables for solving reactive mass transport problems involving chemical dissolution in fluid-saturated porous media with arbitrarily initial porosity distributions. The first advantage

This paper investigates the performance of the XFEM for stiff embedded interfaces and inclusions. It is known that the XFEM may lead to ill-conditioned stiffness matrices and oscillations in the interface traction field, the severity of which depends on the underlying basis functions, as well as on the orientation of the interface. The jumps at the discontinuity are shown to contain quadratic bubble

Superposition-based concurrent multiscale approaches for poromechanical problems aimed at resolving localized fluid flow and deformation of a solid skeleton including its possible fracture are developed. The main departure of the proposed formulation is that the primary unknowns in the coupling zone, the solid displacement u and pore pressure p, are discretized by possibly different numerical models

The interface plays a critical role in the mechanical properties of composites. In the present work, a novel phase-field based cohesive zone model (CZM) is proposed for the cracking simulation. The competition and interaction between the bulk and interfacial cracking are taken into consideration directly in both displacement- and phase-field. A family of modified degradation functions is utilized to

The multiresolution capability provided by the family of Daubechies wavelets is exploited to develop a new computational approach, termed as multiresolution finite wavelet domain method for the fast and hierarchical prediction of transient dynamic problems with focus on wave propagation in one- and two-dimensional structural configurations. In the developed method, both scaling and wavelet functions

Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture

One of the most successful mixed finite element methods in solid mechanics is the enhanced assumed strain (EAS) method developed by Simo and Rifai in 1990. However, one major drawback of EAS elements is the highly mesh dependent accuracy. In fact, it can be shown that not only EAS elements, but every finite element with a symmetric stiffness matrix must either fail the patch test or be sensitive to

Non-Gaussian data assimilation is vital for several applications with nonlinear dynamical systems, including geosciences, socio-economics, infectious disease modeling, and autonomous navigation. Widespread adoption of non-Gaussian data assimilation requires easy-to-implement schemes. We develop, implement, and apply an efficient nonlinear non-Gaussian data assimilation scheme using non-intrusive stochastic

The article is focused on a two-dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These

We present a minimally invasive method for forward propagation of material property uncertainty to full-field quantities of interest in solid dynamics. Full-field uncertainty quantification enables the design of complex systems where quantities of interest, such as failure points, are not known a priori . The method, motivated by the well-known probability density function (PDF) propagation method

Recently, several types of nonlocal discrete differential operators have emerged either from meshfree particle methods or from nonlocal continuum mechanics, such as peridynamics. In this article, we discuss the mathematical formulation as well as construction of the nonlocal discrete differential operators. Based on a least-square minimization procedure and the associated Moore–Penrose inverse, we

Although sensitivity analysis provides valuable information for structural optimization, it is often difficult to use the Hessian in large models since many methods still suffer from inaccuracy, inefficiency, or limitation issues. In this context, we report the theoretical description of a general sensitivity procedure that calculates the diagonal terms of the Hessian matrix by using a new variant

A complex elastoplastic model requires a robust integration procedure of the evolution equations. The performance of the finite element solution is directly affected by the convergence characteristics of the state-update procedure. Thereby, this study proposes a comprehensive numerical integration scheme to deal with generic multi-surface plasticity models. This algorithm is based on the backward Euler

This paper presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian-Eulerian (ALE) formulation to map the wave equation from a time-dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled

Stress analysis is an all-pervasive practice in engineering design. With displacement-based finite element analysis, directly-calculated stress fields are obtained in a post-processing step by computing the gradient of the displacement field–therefore less accurate. In enriched finite element analysis (EFEA), which provides unprecedented versatility by decoupling the finite element mesh from material

Quadrilateral finite elements with five displacements per corner node built up from four triangular elements are developed by a discretization of the Mindlin plate bending theory and based on a modified form of the mixed variational principle for stresses and displacements. The modified principle separates the contribution of shear deformation by taking the transverse shear strains as free functions

Among the various structural optimization tools, topology optimization is the widely used technique in obtaining the initial design of structural components. The resulting topologically optimal initial design will be the input for subsequent structural optimizations such as shape, size and layout optimizations. However, iterative solvers used in conventional topology optimization schemes are known

Polarization-type methods are among the fastest solution methods for FFT-based computational micromechanics. However, their performance depends critically on the choice of the reference material. Only for finitely contrasted materials, optimum-selection strategies are known. This work focuses on adaptive strategies for choosing the reference material, details their efficient implementation, and investigates

In this article, a nonlinear structural formulation that uses a new dimensionless set of generalized displacements is proposed for solving geometrically nonlinear beam problems, being validated by several applications given in literature where a cantilever beam is likely to undergo large displacements. By this approach, useful simplifications and insights are achievable in the analysis process, as

We present a 3D hybrid method which combines the finite element method (FEM) and the spectral boundary integral method (SBIM) to model nonlinear problems in unbounded domains. The flexibility of FEM is used to model the complex, heterogeneous, and nonlinear part— such as the dynamic rupture along a fault with near fault plasticity—and the high accuracy and computational efficiency of SBIM is used to

We propose a new framework for fracture mechanics, based on the idea of an approximate fracture geometry representation combined with approximate interface conditions. Our approach evolves from the shifted interface method, and introduces the concept of an approximate fracture surface composed of the full edges/faces of an underlying grid that are geometrically close to the true fracture geometry.

This article presents the Bézier extraction based isogeometric approach to multi-objective topology optimization of periodic microstructures. The approach utilizes the B-splines based Bézier elements as the finite element representation at both macro and micro levels. The equivalent elastic properties of the representative volume element (RVE) are computed by the numerical homogenization method with

This study proposes a machine learning (ML) based approach for optimizing fiber orientations of variable stiffness carbon fiber reinforced plastic (CFRP) structures, where neural networks are developed to estimate the objective function and analytical sensitivities with respect to design variables as a substitute for finite element analysis (FEA). To reduce the number of training samples and improve

Powell–Sabin B-splines are enjoying an increased use in the analysis of solids and fluids, including fracture propagation. However, the Powell–Sabin B-spline interpolation does not hold the Kronecker delta property and, therefore, the imposition of Dirichlet boundary conditions is not as straightforward as for the standard finite elements. Herein, we discuss the applicability of various approaches

In this work, we present a new monolithic finite element strategy for solving fluid–structure interaction problems involving a compressible fluid and a hyperelastic structure. In the Lagrangian limit, the time-stepping strategy that we propose conserves the total energy, and linear and angular momenta. Detailed proofs with numerical validations are provided. We use a displacement-based Lagrangian formulation

This article presents a consistent computational framework for multiscale first-order finite strain homogenization and stability analyses of rate-independent solids with periodic microstructures. The homogenization formulation is built on a priori discretized microstructure, and algorithms for computing the matrix representations of the homogenized stresses and tangent moduli are consistently derived

In this article, the existing nonsmooth generalized- α method for the simulation of mechanical systems with frictionless contacts, modeled as unilateral constraints, is extended to systems with frictional contacts. On that account, we complement the unilateral constraints with set-valued Coulomb-type friction laws. Moreover, we devise a set of benchmark systems, which can be used to validate numerical

In this study, a parallel framework for the multiscale analysis of three-dimensional lattice structures is developed. The established parallel framework performs the multiscale analysis using the extended multiscale finite element method (EMsFEM) in a parallel environment provided by the portable and extensible toolkit for scientific computation (PETSc), which is a high-performance parallel scientific

We present an adaptation of the multiscale finite element method to the analysis of sandwich beams and plates with complex lattice layers. The proposed modification significantly reduces the number of degrees of freedom (even by four orders) due to the anisotropic higher-order coarse-scale approximation and the novel shape functions that take into account the microscale boundary conditions. Moreover

The use of modern discretization technologies such as hybrid high-order (HHO) methods, coupled with appropriate linear solvers, allow for the robust and fast solution of partial differential equations (PDEs). Although efficient linear solvers have recently been made available for simpler cases, complex geometries remain a challenge for large scale problems. To address this problem, we propose in this

This article presents a proper generalized decomposition (PGD) based Padé approximant for efficient analysis of the stochastic frequency response. Due to the high nonlinearity of the stochastic response with respect to the input uncertainties, the classical stochastic Galerkin (SG) method utilizing polynomial chaos exhibits slow convergence near the resonance. Furthermore, the dimension of the SG method

An efficient approach for topology optimization under uncertainty is presented. Stochastic reduced order models (SROMs) are leveraged for the modeling and propagation of uncertainties within a robust topology optimization (RTO) formulation. The SROM approach provides an alternative to existing uncertainty quantification methods and yields a substantial improvement in efficiency over a classical Monte

A variationally consistent model-based computational homogenization approach for transient chemomechanically coupled problems is developed based on the classical assumption of first-order prolongation of the displacement, chemical potential, and (ion) concentration fields within a representative volume element (RVE). The presence of the chemical potential and the concentration as primary global fields

A new strategy for constructing parametric Galerkin reduced order models is presented in this article. This strategy is achieved thanks to the Riemannian manifold, quotient of the set of full-rank matrices by the orthogonal group. Starting from a set of training parametrized subspaces of the same dimension, namely obtained by the proper orthogonal decomposition, the projection subspace for a new untrained

In the present article the two-dimensional hybrid equilibrium element formulation is initially developed, with quadratic, cubic, and quartic stress fields, for static analysis of compressible and quasi-incompressible elastic solids in the variational framework of the minimum complementary energy principle. Thereafter, the high-order hybrid equilibrium formulation is developed for dynamic analysis of

This work is concerned with computing the effective crack energy of periodic and random media which arises in mathematical homogenization results for the Francfort–Marigo model of brittle fracture. A previous solver based on the fast Fourier transform (FFT) led to solution fields with ringing or checkerboard artifacts and was limited in terms of the achievable accuracy. As computing the effective crack

This paper demonstrates gradient driven density-based topology optimization of transient impact problems with friction. The sensitivities are obtained using the semi-discrete adjoint approach in which the temporal components are obtained by the "differentiate then discretize" whereas the spatial part is determined discretely. In this work, the sensitivity analysis method is extended to a mixed formulation

The current study investigates the stability and failure analysis of thin-walled composite columns with a top-hat and channel cross section, subjected to axial compression. The analyzed structures were made of composite material (carbon fiber reinforced polymer) by autoclave technique. The study included experimental tests on actual specimens and numerical simulations on numerical models using finite

Uncertainties associated with spatially varying parameters are modeled through random fields discretized into a finite number of random variables. Standard discretization methods, such as the Karhunen–Loève expansion, use series representations for which the truncation order is specified a priori. However, when data is used to update random fields through Bayesian inference, a different truncation

This paper considers the problem of determining probabilistic entropy fluctuations, which are important for understanding uncertainty propagation in mechanical systems in the elasto-plastic regime. Probabilistic entropy is conceptualized based on an initial definition by Shannon, which demands discrete representation of the uncertainty source. Numerical analysis is performed using the Response Function

Modeling of roll forming process of sheet metal requires efficient treatment of plastic deformations of thin shells and plates. We suggest a new general approach toward constructing the governing equations of bending of an elastic-plastic Kirchhoff plate on the structural mechanics level. The generic function of the isotropic hardening law is formulated in terms of variables, which are meaningful for

Errors were identified in the article1 subsequent to its publication. Equations (12)–(14) were written incorrectly in the article,1 the corresponding correct equations are ρ ˜ i = ∑ j = 1 N e v j ρ j w ( x j ) ∑ j = 1 N e v j w ( x j ) , (12) w ( x j ) = max 0 , 1 − | | x i − x j | | R fill , (13) ∂ ρ ˜ i ∂ ρ j = v j w ( x j ) ∑ k = 1 N e v k w ( x k ) . (14) These corrections do not change any results

This article presents an explicit topology optimization approach using boundary element method based moving morphable void (MMV). Structural analysis in the conventional MMV approach is performed on a fixed mesh over the entire design domain. However, in the proposed method structural boundaries are described using B-splines and discretized with boundary elements making explicit representation of the

Concurrent atomistic-to-continuum methods commonly employ either a dynamic or a quasi-static continuum model. Both of these approaches have advantages and drawbacks which render their applicability problem-specific. We present a new hybrid static–dynamic continuum model, which combines the advantages of both approaches while potentially removing the drawbacks. This numerical approach is an innovative

The discrete element model (DEM) has attractive advantages in expressing multiple cracks propagation problem in continuum, but the description of material plastic characteristics by current DEM is restricted by the connection model, which is the core procedure in DEM modeling process. A Godunov-type continuum-based DEM model is proposed to solve the dynamic response of materials under high-speed impact

This article proposes an efficient concurrent coupling of two different material scales—a macroscale and a microscale—in a direct solution scheme based on explicit time integration. Both scales may be discretized with different element sizes and the microdomain may exhibit a heterogeneous structure. A surface coupling is described, which imposes the macrovelocities at the interfaces on the microscale

An extended approach is developed by blending isogeometric analysis and the first-order local maximum entropy for the standard and the enhanced fields, respectively. Isogeometric analysis facilitates the accurate parametrization of the geometry in general, particularly the exact geometric parametrization of the conic curves and quadratic surfaces using NURBS. On the other hand, the local maximum entropy

Under compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction

We propose a novel FETI-DP (dual-primal finite element tearing and interconnecting) method for nonlinear variational inequalities. The proposed method is based on a particular Fenchel–Rockafellar dual formulation of the target problem, which yields linear local problems despite the nonlinearity of the target problem. Since local problems are linear, each iteration of the proposed method can be done

A unified contact overlap, termed the Minkowski overlap, between any two shapes is proposed in this paper. This overlap is based on the concept of the Minkowski difference of two shapes, and particularly on the equivalence between the contact state of the two shapes and the location of the origin relative to their Minkowski difference. The Minkowski contact features of a contact, including the overlap

A novel methodology for multiple flaw detection is presented in this study. It combines the dynamic extended finite element method (XFEM) with machine learning for the first time. The extreme learning machine (ELM) is chosen as a learning rule for modeling and prediction. The XFEM is employed to overcome the issues associated with the large quantity of input data required for ELM network training,

We develop a compact fourth-order scheme for the three-dimensional elastic wave equation in frequency space, using the first-order velocity-stress formulation. The scheme is implemented numerically for homogeneous media on a staggered grid, and both the acoustic and elastic cases are considered. We use a one-directional point source of impact, for which Pilant developed a closed solution. Numerical

The analytical approach, which lies at the basis of the exact and natural sciences, rests on the notion that complex systems can be reduced to simple components. This divide-and-conquer approach has shaped contemporary science and technology, and has formed the foundation for virtually all scientific breakthroughs in the 19th and 20th century. It is also the origin of the separation of mechanics into

A total-Lagrangian material point method (TLMPM) with a mixed formulation is proposed in this article for the mechanical analysis of incompressible soft materials coupling the mass growth and massive deformation. In this work, the growth-induced large deformation behavior is handled within the framework of the material point method (MPM) based on the multiplicative decomposition of deformation gradient

Considering structural strength and castability in topology optimization is very attractive from the perspective of theory and application. This article introduces stress considerations into casting-oriented topology optimization to explore structural optimization methods that simultaneously consider the above two vital design indicators. Two stress-related topology optimization formulations for castable

An optimal version of spline dimensional decomposition (SDD) is unveiled for general high-dimensional uncertainty quantification analysis of complex systems subject to independent but otherwise arbitrary probability measures of input random variables. The resulting method involves optimally derived knot vectors of basis splines (B-splines) in some or all coordinate directions, whitening transformation