Linear Elastic Problems Research Articles

Data-Efficient Dimensionality Reduction and Surrogate Modeling of High-Dimensional Stress Fields

Abstract Tensor datatypes representing field variables like stress, displacement, velocity, etc., have increasingly become a common occurrence in data-driven modeling and analysis of simulations. Numerous methods [such as convolutional neural networks (CNNs)] exist to address the meta-modeling of field data from simulations. As the complexity of the simulation increases, so does the cost of acquisition, leading to limited data scenarios. Modeling of tensor datatypes under limited data scenarios remains a hindrance for engineering applications. In this article, we introduce a direct image-to-image modeling framework of convolutional autoencoders enhanced by information bottleneck loss function to tackle the tensor data types with limited data. The information bottleneck method penalizes the nuisance information in the latent space while maximizing relevant information making it robust for limited data scenarios. The entire neural network framework is further combined with robust hyperparameter optimization. We perform numerical studies to compare the predictive performance of the proposed method with a dimensionality reduction-based surrogate modeling framework on a representative linear elastic ellipsoidal void problem with uniaxial loading. The data structure focuses on the low-data regime (fewer than 100 data points) and includes the parameterized geometry of the ellipsoidal void as the input and the predicted stress field as the output. The results of the numerical studies show that the information bottleneck approach yields improved overall accuracy and more precise prediction of the extremes of the stress field. Additionally, an in-depth analysis is carried out to elucidate the information compression behavior of the proposed framework.

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.

Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

A non-intrusive multiscale framework for 2D analysis of local features by GFEM — A thorough parameter investigation

This work comprehensively investigates key parameters associated with a recently proposed non-intrusive coupling strategy for multiscale structural problems. The IGL-GFEMgl combines the Iterative Global Local Method and the Generalized Finite Element Method with global–local enrichment, GFEMgl. Different scales of the problem are solved using distinct finite element codes: the commercial software Abaqus and a research in-house code. An Iterative Global–Local non-intrusive algorithm is employed to couple the solutions provided by the two solvers, with the process accelerated by Aitken’s relaxation. Slight modifications have been introduced, and the resulting accuracy and computational performance are discussed using numerical examples. The problems investigated explore the coupling strategy within the context of 2D linear elastic problems, which include voids and crack propagation described at the local scale solved by the in-house code. A noteworthy trade-off between reducing iterations and increasing the time to solve the local problems is observed. Despite the high accuracy achieved, the two versions of the coupling strategy, namely the monolithic and staggered algorithms, exhibit different computational performances when the GFEMgl parameters, such as the number of global–local cycles and the size of the buffer zone, are evaluated for the crack propagation simulation.

Mixed displacement–pressure formulations and suitable finite elements for multimaterial problems with compressible and incompressible models

Multimaterial problems in linear and nonlinear elasticity are some of the least explored using mixed finite element formulations with higher-order elements. The fundamental issue in adapting the mixed displacement–pressure formulations with linear and higher-order continuous elements for the pressure field is their inability to capture pressure and stress jumps across material interfaces. In this paper, for the first time in literature, we perform comprehensive studies of multimaterial problems in elasticity consisting of compressible and incompressible material models using the mixed displacement–pressure formulation to assess the performance of different element types in accurately resolving pressure fields within the domains and pressure jumps across material interfaces. In particular, inf–sup stable displacement–pressure combinations with element-wise discontinuous pressure for triangular and tetrahedral elements are considered and their performance is assessed along with the Q1/P0 element and Taylor–Hood elements using several numerical examples. The results show that Taylor–Hood elements fail to capture the stress jumps due to the continuity of DOFs across elements, the Crouzeix–Raviart (P2b/P1dc) element yields substantially poor pressure fields despite a significant increase in pressure degrees of freedom and that the P3/P1dc element produces superior quality results fields when compared with the P2b/P1dc element.

Solving linear elasticity benchmark problems via the overset improved element-free Galerkin-finite element method

A novel approach for the solution of linear elasticity problems is introduced in this communication, which uses a hybrid chimera-type technique based on both finite element and improved element-free Galerkin methods. The proposed overset improved element-free Galerkin-finite element method (Ov-IEFG-FEM) for linear elasticity uses the finite element method (FEM) throughout the entire problem geometry, whereas a fine distribution of overlapping nodes is used to perform higher order approximations via the improved element-free Galerkin (IEFG) technique in regions demanding more computational accuracy. The method relies on keeping the FEM-based results in those regions where low order of approximation is enough to provide the required accuracy, i.e. outside the region where the solution will be enriched via the IEFG technique. The overlapping domains perform an iterative transfer of kinematics information through well-defined immersed boundaries, and a detailed explanation on this regard is also presented in this communication. The Ov-IEFG-FEM is used in a set of increasingly complex linear elasticity problems, and the outcomes demonstrate the suitability and reliability of this technique to solve such problems in an accurate and remarkably simple manner.

Approximation of acoustic black holes with finite element mixed formulations and artificial neural network correction terms

Wave propagation in elastodynamic problems in solids often requires fine computational meshes. In this work we propose to combine stabilized finite element methods (FEM) with an artificial neural network (ANN) correction term to solve such problems on coarse meshes. Irreducible and mixed velocity–stress formulations for the linear elasticity problem in the frequency domain are first presented and discretized using a variational multiscale FEM. A non-linear ANN correction term is then designed to be added to the FEM algebraic matrix system and produce accurate solutions when solving elastodynamics on coarse meshes. As a case study we consider acoustic black holes (ABHs) on structural elements with high aspect ratios such as beams and plates. ABHs are traps for flexural waves based on reducing the structural thickness according to a power-law profile at the end of a beam, or within a two-dimensional circular indentation in a plate. For the ABH to function properly, the thickness at the termination/center must be very small, which demands very fine computational meshes. The proposed strategy combining the stabilized FEM with the ANN correction allows us to accurately simulate the response of ABHs on coarse meshes for values of the ABH order and residual thickness outside the training test, as well as for different excitation frequencies.

A locking-free mixed enriched Galerkin method of arbitrary order for linear elasticity using the stress–displacement formulation

In this paper, we develop an arbitrary-order locking-free enriched Galerkin method for the linear elasticity problem using the stress–displacement formulation in both two and three dimensions. The method is based on the mixed discontinuous Galerkin method in Wang et al. (2020), but with a different stress approximation space that enriches the arbitrary order continuous Galerkin space with some piecewise symmetric-matrix valued polynomials. The enriched Galerkin method maintains locking-free property of the mixed discontinuous Galerkin method in Wang et al. (2020), but has fewer degrees of freedom for commonly used and appropriately high orders. We present some numerical examples in two and three dimensions to demonstrate the effectiveness of the proposed method.

Dimension reduction and homogenization of composite plate with matrix pre-strain

This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic perforated plate made of stiff material with holes filled by a soft matrix material. The structure is clamped on a cylindrical part. Two cases of asymptotic analysis are considered: one without pre-strain and the other with matrix pre-strain. In both cases, the total elastic energy is in the von-Kármán (vK) regime ( ε 5 ). A new splitting of the displacements is introduced to analyze the asymptotic behavior. The displacements are decomposed using the Kirchhoff–Love (KL) plate displacement decomposition. The use of a re-scaling unfolding operator allows for deriving the asymptotic behavior of the Green St. Venant’s strain tensor in terms of displacements. The limit homogenized energy is shown to be of vK type with linear elastic cell problems, established using the Γ-convergence. Additionally, it is shown that for isotropic homogenized material, our limit vK plate is orthotropic. The derived results have practical applications in the design and analysis of composite structures.

A New Mixed Finite Element for the Linear Elasticity Problem in 3D

A New Mixed Finite Element for the Linear Elasticity Problem in 3D

Point collocation with mollified piecewise polynomial approximants for high‐order partial differential equations

AbstractThe solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of cell‐wise defined piecewise polynomials with a smooth mollifier of certain characteristics. The properties of the mollified basis functions are governed by the order of the piecewise functions and the smoothness of the mollifier. In this work, we exploit the high‐order and high‐smoothness properties of the mollified basis functions for solving PDEs through the point collocation method. The basis functions are evaluated at a set of collocation points in the domain. In addition, boundary conditions are imposed at a set of boundary collocation points distributed over the domain boundaries. To ensure the stability of the resulting linear system of equations, the number of collocation points is set larger than the total number of basis functions. The resulting linear system is overdetermined and is solved using the least square technique. The presented numerical examples confirm the convergence of the proposed approximation scheme for Poisson, linear elasticity, and biharmonic problems. We study in particular the influence of the mollifier and the spatial distribution of the collocation points.

A reduced order variational spectral method for efficient construction of eigenstrain‐based reduced order homogenization models

AbstractReduced order models (ROMs) are often coupled with concurrent multiscale simulations to mitigate the computational cost of nonlinear computational homogenization methods. Construction (or training) of ROMs typically requires evaluation of a series of linear or nonlinear equilibrium problems, which itself could be a computationally very expensive process. In the eigenstrain‐based reduced order homogenization method (EHM), a series of linear elastic microscale equilibrium problems are solved to compute the localization and interaction tensors that are in turn used in the evaluation of the reduced order multiscale system. These microscale equilibrium problems are typically solved using either the finite element method or semi‐analytical methods. In the present study, a reduced order variational spectral method is developed for efficient computation of the localization and interaction tensors. The proposed method leads to a small stiffness matrix that scales with the order of the reduced basis rather than the number of degrees of freedom in the finite element mesh. The reduced order variational spectral method maintains high accuracy in the computed response fields. A speedup higher than an order of magnitude can be achieved compared to the finite element method in polycrystalline microstructures. The accuracy and scalability of the method for large polycrystals and increasing phase property contrast are investigated.

Analytic solution for two dimensional beam problems: Pure displacement boundary conditions

The present manuscript delineates the derivation of strong solutions for the linear elasticity problem in a two dimensional rectangular beam. The materials under consideration can exhibit either isotropic or orthotropic properties. Additionally, the analysis is not restricted to slender beams because the ratio between the length and the height of the beam can be arbitrary. The boundary conditions fall into the Dirichlet category, implying that both horizontal and vertical displacements are specified on all four surfaces. The sole requirement is that these surface displacements are continuous functions, although they may not necessarily be smooth. Since the displacements at the surfaces can be arbitrary, there is no need to consider approximations, such as those concerning local (small) boundaries in slender beams. Nonetheless, it is demonstrated that an equivalent principle to the Saint-Venant principle exists for pure displacement boundary conditions. The partial differential equations are solved through the separation of variables method, leading to the identification of two solution types, encompassing both cosine and sine Fourier series. Particular emphasis is placed on evaluating the convergence of these solutions. For two distinct and general examples, it is confirmed that the solutions indeed exist.

Impedance operator of a curved thin layer in linear elasticity with voids

We consider a two‐dimensional transmission problem of linear elasticity with voids in a fixed domain coated by a curved thin layer . Our aim is to model the effect of the thin layer on the fixed domain by an impedance boundary condition. For that, we use the techniques of asymptotic expansion to approximate the transmission problem by an impedance problem set in the fixed domain , and we prove an error estimate for the approximate impedance problem.

Rational generalized mixed finite element method for 2D linear elastic problems

Combining the ideas of rational displacement finite element and generalized mixed finite element, the rational generalized mixed element with 5 nodes is proposed. The finite formulations for the proposed element are derived on the basis of the generalized mixed variational principle, and the two field variables of displacement and stress can be solved simultaneously. Fundamental solutions of planar linear elasticity are adopted as the displacement interpolation function, while polynomials satisfying the equilibrium equation are adopted as the stress interpolation function. Both the displacement and stress boundary conditions can be introduced directly at same time in the proposed method, which contributes to being a more rational model. The solution technique is utilized to analyze a couple of representative static problems, and the results show the properties of excellent accuracy and good convergence. In addition, it is found to be evident for improvement of the accuracy when the stress boundary conditions are applied.

Closed-form solutions for axially non-uniform Timoshenko beams and frames under static loading

This paper presents the Green’s Functions Stiffness Method (GFSM) for solving linear elastic static problems in arbitrary axially non-uniform Timoshenko beams and frames subjected to general external loads and bending moments. The GFSM is a mesh reduction method that seamlessly integrates elements from the Stiffness Method (SM), Finite Element Method (FEM), and Green’s Functions (GFs), resulting in a highly versatile methodology for structural analysis. It incorporates fundamental concepts such as stiffness matrices, shape functions, and fixed-end forces, in line with SM and FEM frameworks. Leveraging the capabilities of GFs, the method facilitates the derivation of closed-form solutions, addressing a gap in existing methods for analyzing non-uniform reticular structures which are typically limited to simple cases like single-span beams with specific axial variations and loading scenarios. The effectiveness of the GFSM is demonstrated through three practical examples, showcasing its applicability in analyzing non-uniform beams and plane frames, thereby broadening the scope of closed-form solutions for axially non-uniform Timoshenko structures.

Linel2D-Net: A deep learning approach to solving 2D linear elastic boundary value problems on image domains

Efficient solution of physical boundary value problems (BVPs) remains a challenging task demanded in many applications. Conventional numerical methods require time-consuming domain discretization and solving techniques that have limited throughput capabilities. Here, we present an efficient data-driven DNN approach to non-iterative solving arbitrary 2D linear elastic BVPs. Our results show that a U-Net-based surrogate model trained on a representative set of reference FDM solutions can accurately emulate linear elastic material behavior with manifold applications in deformable modeling and simulation.

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

Solution of planar elastic stress problems using stress basis functions

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ p , is any stress in equilibrium with the loading. The second part, σ h is a self-equilibrated stress field on the unloaded body. In both methods, σ h is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.