Linear Elastic Problems Research Articles

Modeling Issues in Problems of the Elasticity and Viscoelasticity Theory

The development of computer technology and information technology does not exclude preliminary analytical modeling in complex problems of mechanics. The approach proposed in this work allows one to obtain reasonable simplified equations that admit analytical or numerical-analytical solutions. Authors research are devoted to method elaboration for space problems of viscoelasticity. The approach for solving problems of nonlinear elasticity theory, when final deformations or physical nonlinear properties of materials are taken into consideration, is proposed. New perturbation method for solving of nonlinear equations in particular derivatives is suggested. Such approach is allowed to reduce the solution of complicated problems of linear elasticity and viscoelasticity to subsequently solved boundary problems of potential theory. New linear problems are investigated, in particular, problems on load transference from stringer to single layer or multilayer solids, on stress-strain state of fibrous composite with crack in matrix (plate and axisymmetric problems), numerous contact problems (pressing of hard stamp in an orthotropic plate with different anisotropy). Problems of nonlinear elasticity theory on stress-strain state of a plate with a circular hole under different types of loading are solved.

Fast multiscale contrast independent preconditioners for linear elastic topology optimization problems

The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process is iterative, and the gradients are obtained by an adjoint analysis, which requires the numerical solution of large high-contrast linear elastic problems with features spanning several length scales. The size of the discretized problems forces the utilization of iterative linear solvers with solution time dependent on the quality of the preconditioner. The lack of clear separation between the scales, as well as the high-contrast, imposes severe challenges on the standard preconditioning techniques. Thus, here we propose new methods for the high-contrast elasticity equation with performance independent of the high-contrast and the multi-scale structure of the elasticity problem. The solvers are based on two-levels domain decomposition techniques with a carefully constructed coarse level to deal with the high-contrast and multi-scale nature of the problem. The construction utilizes spectral equivalence between scalar diffusion and each displacement block of the elasticity problems and, in contrast to previous solutions proposed in the literature, is able to select the appropriate dimension of the coarse space automatically. The new methods inherit the advantages of domain decomposition techniques, such as easy parallelization and scalability. The presented numerical experiments demonstrate the excellent performance of the proposed methods.

Approximation of the axisymmetric elasticity equations

In this article we consider the linear elasticity problem in an axisymmetric three dimensional domain, with data which are axisymmetric and have zero angular component. The weak formulation of the three dimensional problem reduces to a two dimensional problem on the meridian domain, involving weighted integrals. The problem is formulated in a mixed method framework with both the stress and displacement treated as unknowns. The symmetry condition for the stress tensor is weakly imposed. Well posedness of the continuous weak formulation and its discretization are shown. Two approximation spaces are discussed and corresponding numerical computations are presented.

Local elastic moduli of simple random composites computed at different length scales

Techniques like nanoindentation and atomic force microscopy can estimate the local elastic moduli in a region surrounding the probe used. For composites with phase regions much larger than the size of the probe, these procedures can identify the phases via their different elastic moduli but identifying phase regions that are on the same size scale as the indent is more problematic. This paper looks at three random 3D 8003 voxel composite models, each consisting of a matrix and spherical inclusions. One model has non-overlapping spheres and two models have overlapping spheres, with two and three distinct phases. The linear elastic problem is solved for each microstructure, and histograms are made of the local Young’s moduli over a number of sub-volumes (SVs), averaged over progressively larger SVs. The number and shape of histogram peaks change from N delta functions, where N is the number of elastically distinct phases, at the 1 voxel SV limit, to a single delta function located at the value of the effective global Young’s modulus, when the SV equals the unit cell volume. The phase volume fractions are also tracked for each bin in the Young’s modulus histograms, showing the phase make-up of bin in the histogram. There are clear differences seen between the non-overlapping and three-phase overlapping models and the two-phase overlapping sphere model, because of different size microstructural features, characterized by the average value of size as computed by the W(q) function. In the three-phase model, a peak that is originally all phase 3 persists at its same location, but as the size of the SVs increase, it is made up of a mixture of phases, so that it cannot be identified with a single phase even though it remains a clear peak. These results give some guidance as to what probe size might be useful in distinguishing different phases by local elastic moduli measurements, and how the length scales of the probe and the microstructure interact.

Features of the stress-strain state in the corner zones of structures

Corner regions of structures are characterized by a complex stress-strain state (SSS), caused by both the shape of the boundary or the “geometric factor”, and the finite rupture of the specified forced deformations, mechanical properties that go to the contact surface of the elements of composite structures. The relevance of the work consists in the analysis of local SSS in the corner zone of the region boundary with various study approaches: theoretical, experimental. For the theoretical analysis of the SSS of the region, we consider the resolving system of equations of the problem of the theory of elasticity in the zone of the cutout of the boundary and elements of the similarity theory; the photoelasticity method is used for experimental studies. An experimental solution was obtained on polymer composite models with an angle cut of the boundary under the action of forced deformations in one of the regions. The analysis of possibilities of theoretical, experimental approaches of SSS studies in zones with irregular point of border allows to reveal zones of applicability of solutions of linear elasticity problem.

A GPU-based caching strategy for multi-material linear elastic FEM on regular grids

In this study, we present a novel strategy to the method of finite elements (FEM) of linear elastic problems of very high resolution on graphic processing units (GPU). The approach exploits regularities in the system matrix that occur in regular hexahedral grids to achieve cache-friendly matrix-free FEM. The node-by-node method lies in the class of block-iterative Gauss-Seidel multigrid solvers. Our method significantly improves convergence times in cases where an ordered distribution of distinct materials is present in the dataset. The method was evaluated on three real world datasets: An aluminum-silicon (AlSi) alloy and a dual phase steel material sample, both captured by scanning electron tomography, and a clinical computed tomography (CT) scan of a tibia. The caching scheme leads to a speed-up factor of ×2-×4 compared to the same code without the caching scheme. Additionally, it facilitates the computation of high-resolution problems that cannot be computed otherwise due to memory consumption.

A second‐order isoparametric element method to solve plane linear elastic problem

AbstractConsidering the both effect of boundary approximation and numerical quadrature, a second‐order isoparametric element method is given to solve the homogeneous isotropic plane linear elasticity problem in domain Ω with curved boundary. By using technically analysis, the optimal error estimate with is obtained, where the function is an extension of the true solution to . It yields better accuracy than traditional quadratic finite element method. Finally, two numerical examples are presented, which further illustrate the analytical result and show the scheme is effective.

Optimality criteria method in 2D linearized elasticity problems

In optimal design problems the goal is to find an optimal distribution of materials in a fixed domain which optimizes a criterion, computed through the solution of a partial differential equation modelling the involved physics. We consider optimal design problems in the setting of linearized elasticity, and restrict ourselves to domains filled with two isotropic elastic phases. Since a classical solution usually does not exist, we use the homogenization method in order to get a proper relaxation of the original problem. Unfortunately, the related G-closure problem in linearized elasticity is not solved. However, in the compliance minimization problem, G-closure can be replaced by a smaller subset made of sequential laminates, which is explicitly known. Moreover, for the compliance minimization the necessary conditions of optimality are easily derived, which enables a development of optimality criteria method for the numerical solution. We explicitly calculate the lower Hashin-Shtrikman bound on the complementary energy in two space dimensions, and necessary conditions of optimality. We develop an optimality criteria method for the single state compliance minimization problems.

A simple Galerkin meshless method, the Fragile Points method using point stiffness matrices, for 2D linear elastic problems in complex domains with crack and rupture propagation

SummaryThe Fragile Points method (FPM) is an elementarily simple Galerkin meshless method, employing Point‐based discontinuous trial and test functions only, without using element‐based trial and test functions. In this study, the algorithmic formulations of FPM for linear elasticity are given in detail, by exploring the concepts of point stiffness matrices and numerical flux corrections. Advantages of FPM for simulating the deformations of complex structures, and for simulating complex crack propagations and rupture developments, are also thoroughly discussed. Numerical examples of deformation and stress analyses of benchmark problems, as well as of realistic structures with complex geometries, demonstrate the accuracy, efficiency, and robustness of the proposed FPM. Simulations of crack initiation and propagations are also given in this study, demonstrating the advantages of the present FPM in modeling complex rupture and fracture phenomena. The crack and rupture propagation modeling in FPM is achieved without remeshing or augmenting the trial functions as in standard, extended, or generalized finite element method. The simulation of impact, penetration, and other extreme problems by FPM will be discussed in our future articles.

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

Polygonal finite elements offer an increased freedom in terms of mesh generation at the price of more complex, often rational, shape functions. Thus, the numerical integration of rational interpolants over polygonal domains is one of the challenges that needs to be solved. If, additionally, strong discontinuities are present in the integrand, e.g., when employing fictitious domain methods, special integration procedures must be developed. Therefore, we propose to extend the conventional quadtree-decomposition-based integration approach by image compression techniques. In this context, our focus is on unfitted polygonal elements using Wachspress shape functions. In order to assess the performance of the novel integration scheme, we investigate the integration error and the compression rate being related to the reduction in integration points. To this end, the area and the stiffness matrix of a single element are computed using different formulations of the shape functions, i.e., global and local, and partitioning schemes. Finally, the performance of the proposed integration scheme is evaluated by investigating two problems of linear elasticity.

A symplectic analytical singular element for V-notched analyses in elastic and viscoelastic plane problems

The linear elastic and viscoelastic problems with V-shaped notches under in-plane loading are investigated in this paper. We consider two different sets of boundary conditions on the radial edges: Free-Free case and Clamped-Clamped case. A novel singular finite element (SASE) with arbitrary high-order accuracy for the notched problems is proposed. Firstly, via Laplace transform, the original viscoelastic problem is transformed into a corresponding elastic one. Then we construct the SASE by using elastic symplectic eigen solutions with higher order expanding terms. The SASE can depict the characteristics of displacement fields and singular stress fields in the vicinity of the notch vertex having arbitrary opening angle. By taking advantage of the symplectic eigen solutions in the SASE, Mode I and/or Mode II notch stress intensity factors can be determined directly without any post-processing. Fine finite element meshes are not required. Numerical examples are provided to illustrate the validity and accuracy of the present method.

A simple, first-order, well-conditioned, and optimally convergent Generalized/eXtended FEM for two- and three-dimensional linear elastic fracture mechanics

This paper proposes a first-order Generalized/eXtended Finite Element Method (G/XFEM) for 2-D and 3-D linear elastic fracture mechanics problems. The conditioning of the method is of the same order as the standard FEM and it is robust with respect to the position of the mesh relative to 2-D or 3-D fractures. The method achieves an optimal rate of convergence in energy norm even when the solution of the problem in the neighborhood of the fracture front is not contained in the spaces spanned by the adopted singular enrichments. Control of conditioning associated with Heaviside enrichment functions is achieved by shifting them by their nodal values, combined with a diagonal pre-conditioner. In the case of singular enrichment functions, a simple extension of the enrichment shifting concept, denoted as discontinuous-shifting, is adopted. These enrichment modifications are space-preserving—the solution space spanned by the modified enrichments is the same as the space spanned by the unmodified ones. This is often not the case for conditioning control strategies proposed in the literature. The computational cost of these enrichment modifications is negligible and their implementation in existing G/XFEM software is straightforward. The optimal convergence, well-conditioning, and robustness of the method are numerically illustrated with the aid of representative 2-D and 3-D problems, including the case of a non-planar fracture with a curved fracture front.

Active set expansion strategies in MPRGP algorithm

The paper investigates strategies for expansion of active set that can be employed by the MPRGP algorithm. The standard MPRGP expansion uses a projected line search in the free gradient direction with a fixed step length. Such a scheme is often too slow to identify the active set, requiring a large number of expansions. We propose to use adaptive step lengths based on the current gradient, which guarantees the decrease of the unconstrained cost function with different gradient-based search directions. Moreover, we also propose expanding the active set by projecting the optimal step for the unconstrained minimization. Numerical experiments demonstrate the benefits (up to 78% decrease in the number of Hessian multiplications) of our expansion step modifications on two benchmarks – contact problem of linear elasticity solved by TFETI and machine learning problems of SVM type, both implemented in PERMON toolbox.

Meshfree numerical approach based on the Finite Pointset Method for static linear elasticity problems

In this work a promising meshfree numerical approach is extended and implemented for the first time in the field of linear static elasticity. A real meshfree method, the so called Finite Pointset Method (FPM), is applied and implemented in order to solve the strong/classical form of the governing partial differential equations (PDEs), the well known Navier–Cauchy equations. FPM numerical performance is demonstrated for several classical and well known studied benchmark problems and comparison with other approaches is also reported. Numerical results suggest that FPM method is stable with good convergence behavior for 2D linear static elasticity problems.

Higher-order mesh generation using CISAMR: A case study on bias in presentation and interpretation of results

A modified version of the Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR) algorithm is presented for the construction of higher-order Lagrangian and NURBS-enhanced (NE) finite element meshes. CISAMR non-iteratively transforms a structured grid into a conforming mesh with an upper bound of three on elements aspect ratio. In this work, we introduce new algorithmic aspects for generating higher-order Lagrangian and NE meshes using CISAMR. For each type of element, a comprehensive study is then provided on the performance of first, second, and third-order meshes for solving linear elastic problems with smooth and oscillatory curvilinear edges. Local gradient recovery error and computational cost are used as performance metrics in these examples. Outcomes are then used as a case study to elucidate common sources of bias in presenting and interpreting results. In particular, we show how bias, whether purposeful or not, could lead to misleading conclusions regarding the performance of a method.

A hybridizable discontinuous Galerkin Generalized Multiscale Finite element method for highly heterogeneous linear elasticity problems

In this paper, we consider a multiscale hybridizable discontinuous Galerkin (HDG) for heterogeneous linear elasticity problem in a mixed formulation. Within the framework of HDG, the entire problem can be decomposed into a global problem defined in coarse interfaces and several local problems in coarse elements. Our goal here is to propose a multiscale basis space defined in the coarse interface. We will employ local snapshot spaces and local spectral decomposition following the concept of Generalized Multiscale Finite Element Methods (GMsFEM). Numerical experiments on highly heterogeneous material and nearly incompressible material are provided to verify the efficiency of the proposed method.

A review on development of rapid numerical quadrature in finite element method

Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally requires the solution to boundary value problems for partial differential equations. In particular, the focus will be on the Finite Element Method and its application to linear and nonlinear problems. Aim of the present proposal is to: initiate the investigation on the Universal Matrix method which is introduced presently for linear elastic and transient heat transfer problems when applied to non-linear elastic and inelastic problems. Multiple databases are searched for literature and limiting to last ten years. The keywords selected for the search were a combination of the nonlinear heat transient, inelastic problems. Universal Matrix Method applied to nonlinear heat transient & inelastic problems which a time is reducing than the existing methods like Gauss Quadrature methods, etc. Research on usage of various methods other than Gauss Quadrature method applied to linear problems extended to nonlinear heat transient & inelastic problems.

Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element

This paper uses a locking-free nonconforming Crouzeix–Raviart finite element to solve the planar linear elastic eigenvalue problem with homogeneous pure displacement boundary condition. Making full use of the Poincaré inequality, we obtain the guaranteed lower bounds for eigenvalues. Besides, we also use the nonconforming Crouzeix–Raviart finite element to the planar linear elastic eigenvalue problem with the pure traction boundary condition, and obtain the guaranteed lower eigenvalue bounds. Finally, numerical experiments with MATLAB program are reported.

An extended stochastic pseudo‐spectral Galerkin finite element method (XS‐PS‐GFEM) for elliptic equations with hybrid uncertainties

SummaryNouy and Clement introduced the stochastic extended finite element method to solve linear elasticity problem defined on random domain. The material properties and boundary conditions were assumed to be deterministic. In this work, we extend this framework to account for multiple independent input uncertainties, namely, material, geometry, and external force uncertainties. The stochastic field is represented using the polynomial chaos expansion. The challenge in numerical integration over multidimensional probabilistic space is addressed using the pseudo‐spectral Galerkin method. Thereafter, a sensitivity analysis based on Sobol indices using the derived stochastic extended Finite Element Method solution is presented. The efficiency and accuracy of the proposed novel framework against conventional Monte Carlo methods is elucidated in detail for a few one and two dimensional problems.

A dual hybrid virtual element method for plane elasticity problems

A dual hybrid Virtual Element scheme for plane linear elastic problems is presented and analysed. In particular, stability and convergence results have been established. The method, which is first order convergent, has been numerically tested on two benchmarks with closed form solution, and on a typical microelectromechanical system. The numerical outcomes have proved that the dual hybrid scheme represents a valid alternative to the more classical low-order displacement-based Virtual Element Method.