Journal of Multiscale Modelling

In this work, a method for the application of multi-fidelity modeling to the reliability analysis of 2D elastostatic structures using the boundary element method (BEM) is proposed. Reliability analyses were carried out on a rectangular plate with a center circular hole subjected to uniaxial tension using Monte Carlo simulations (MCS), the first-order reliability method (FORM), and the second-order reliability method (SORM). Two BEM models were investigated, a low-fidelity model (LFM) of 20 elements and a high-fidelity model (HFM) of 100 elements. The response of these models at several design points was used to create multi-fidelity models (MFMs) utilizing second-order polynomial response surfaces and their reliability, alongside that of the LFM and the HFM, was evaluated. Results show that the MFMs that directly called the LFM were significantly superior in terms of accuracy to the LFM, achieving very similar levels of accuracy to the HFM, while also being of similar computational cost to the LFM. These direct MFMs were found to provide good substitutes for the HFM for MCS, FORM, and SORM.

The computational implementation of moving least squares (MLS) shape functions is an important step to consider in some versions of the meshless local Petrov–Galerkin (MLPG) method for a variety of two-dimensional engineering problems. Here, the usage of conventional Gaussian quadrature in the MLPG may require an excessive number integration points to achieve acceptable accuracy. In addition, since for each integration point a search for nearby points contributing to the construction of the MLS shape functions is required, considerable increase in computational cost is often observed. Herein, an efficient hybrid implementation of CPU and GPU is proposed to accelerate the construction of MLS shape functions for MLPG. To this end, a new K-d-Tree (K-dimensional tree)-based data structure is introduced in order to accelerate the calculations involving computational geometry formulas such as MLS. The results are compared for implementations in CPU and $\mathrm{\text{CPU}}+\mathrm{\text{GPU}}$ using K-d-Tree with the traditional algorithm of brute force search of the neighboring points when dealing with two-dimensional linear elasticity problems. Finally, the performance gain in the numerical approximation is obtained when using the highlighted K-d-Tree.

A three-dimensional (3D) boundary element method for small strains crystal plasticity is described. The method, developed for polycrystalline aggregates, makes use of a set of boundary integral equations for modeling the individual grains, which are represented as anisotropic elasto-plastic domains. Crystal plasticity is modeled using an initial strains boundary integral approach. The integration of strongly singular volume integrals in the anisotropic elasto-plastic grain-boundary equations are discussed. Voronoi-tessellation micro-morphologies are discretized using nonstructured boundary and volume meshes. A grain-boundary incremental/iterative algorithm, with rate-dependent flow and hardening rules, is developed and discussed. The method has been assessed through several numerical simulations, which confirm robustness and accuracy.

Multiscale analyses considering the stretching problem in plates composed of metal matrix composites (MMC) have been performed using a coupled BEM/FEM model, where the boundary element method (BEM) and the finite element method (FEM) models, respectively, the macrocontinuum and the material microstructure, denoted as representative volume element (RVE). The RVE matrix zone behavior is governed by the von Mises elasto-plastic model while elastic inclusions have been incorporated to the matrix to improve the material mechanical properties. To simulate the microcracks evolution at the interface zone surrounding the inclusions, a modified cohesive fracture model has been adopted, where the interface zone is modeled by means of cohesive contact finite elements to capture the effects of phase debonding. Thus, this paper investigates how this phase debonding affects the microstructure mechanical behavior and consequently affects the macrostructure response in a multiscale analysis. For that, initially, only RVEs subjected to a generic strain are analyzed. Then, multiscale analyses of plates have been performed being each macro point represented by a RVE where the macro-strain must be imposed to solve its equilibrium problem and obtain the macroscopic constitutive response given by the homogenized values of stress and constitutive tensor fields over the RVE.

In this work, a multi-scaling homogenization process using boundary element formulation (BEM) for modeling a two-dimensional multi-phase microstructure containing irregular’s inclusions is presented. The BEM is very attractive for multiscale modeling tools for heterogeneous materials. In this approach, the iterative inhomogeneity discretization of the external boundary is disregarded, leading to a computational low cost. This approach was used for solving the elastic problem of a representative volume element (RVE) and the field theory medium. The main goal relies on finding the effective properties of micro-heterogeneous materials within a homogeneous and orthotropic matrix. Expressions for evaluating the effective properties under Plane Stress (PT) for orthotropic materials were also presented. Generally, the numerical models consider the graphite nodules as voids for GGG-40 and the roundness is close circular geometry. In this sense, a nodular cast iron GGG-40 microgram was obtained by X-ray computed tomography and Laser Confocal Microscope System, allowing the modeling of the true nodule shape. The numerical results showed good agreement with the experimental tests. The inclusions of graphite were considered as voids in the material matrix. Experimental stress–strain tests and micrographic analysis were used to determine the Young’s modulus, spatial distributions, as well as, nodule shape. The numerical in this work was compared with the obtained experimental results in this work. The comparison between the obtained experimental data with those available in the literature also showed good agreement.

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

This paper presents a spectral boundary element formulation for analysis of structures subjected to dynamic loading. Two types of spectral elements based on Lobatto polynomials and Legendre polynomials are used. Two-dimensional analyses of elastic wave propagation in solids with and without cracks are carried out in the Laplace frequency domain with both conventional BEM and the spectral BEM. By imposing the requirement of the same level of accuracy, it was found that the use of spectral elements, compared with conventional quadratic elements, reduced the total number of nodes required for modeling high-frequency wave propagation. Benchmark examples included a simple one-dimensional bar for which analytical solution is available and a more complex crack problem where stress intensity factors were evaluated. Special crack tip elements are developed for the first time for the spectral elements to accurately model the crack tip fields. Although more integration points were used for the integrals associated with spectral elements than the conventional quadratic elements, shorter computation times were achieved through the application of the spectral BEM. This indicates that the spectral BEM is a more efficient method for the numerical modeling of structural health monitoring (SHM) processes, in which high-frequency waves are commonly used to detect damage, such as cracks, in structures.

A recently proposed criterion is used to study the behavior of debonds produced at a fiber–matrix interface. The criterion is based on the Linear Elastic–(Perfectly) Brittle Interface Model (LEBIM) combined with a Finite Fracture Mechanics (FFM) approach, where the stress and energy criteria are suitably coupled. Special attention is given to the discussion about the symmetry of the debond onset and growth in an isolated single fiber specimen under uniaxial transverse tension. A common composite material system, glass fiber–epoxy matrix, is considered. The present methodology uses a two-dimensional (2D) Boundary Element Method (BEM) code to carry out the analysis of interface failure. The present results show that a non-symmetrical interface crack configuration (debonds at one side only) is produced by a lower critical remote load than the symmetrical case (debonds at both sides). Thus, the non-symmetrical solution is the preferred one, which agrees with the experimental evidences found in the literature.

This paper presents an advanced formulation of the time-domain two-dimensional (2D) boundary element method (BEM) for an elastic, homogeneous unsaturated soil subjected to dynamic loadings. Unlike the usual time-domain BEM, the present formulation applies a convolution quadrature which requires only the Laplace-domain instead of the time-domain fundamental solutions. The coupled equations governing the dynamic behavior of unsaturated soils ignoring contributions of the inertia effects of the fluids (water and air) are derived based on the poromechanics theory within the framework of a suction-based mathematical model. In this formulation, the solid skeleton displacements $u_i$, water pressure $p_w$ and air pressure $p_a$ are presumed to be independent variables. The fundamental solutions in Laplace transformed-domain for such a dynamic $u-p_w-p_a$ theory have been obtained previously by authors. Then, the BE formulation in time is derived after regularization by partial integrations and time and spatial discretizations. Thereafter, the BE formulation is implemented in a 2D boundary element code (PORO-BEM) for the numerical solution. To verify the accuracy of this implementation, the displacement response obtained by the boundary element formulation is verified by comparison with the elastodynamics problem.