Nonlinear Solid Mechanics Problems Research Articles

A robust radial point interpolation method empowered with neural network solvers (RPIM-NNS) for nonlinear solid mechanics

In this work, we proposed a robust radial point interpolation method empowered with neural network solvers (RPIM-NNS) for solving highly nonlinear solid mechanics problems. It is enabled by neural network solvers via minimizing an energy-based functional loss. The RPIM-NNS has the following key ingredients: (1) It uses radial basis functions (RBFs) for displacement interpolation at arbitrary points in the problem domain, permitting irregular node distributions. (2) Nodes are placed also beyond the domain boundary, allowing the convenient implementation of boundary conditions of both Dirichlet and Neumann types. (3) It uses strain energy in an integral form as a part of the loss function, ensuring solution stability. (4) A well-developed gradient descendant algorithm in machine learning is employed to find the optimal solution, enabling robustness and ease in handling material and geometrical nonlinearities. (5) The proposed RPIM-NNS is compatible with parallel computing schemes. The performance of this method is tested using nonlinear problems including Cook's membrane and 3D twisting rubber problems, demonstrating its remarkable stability and robustness. This work, which seamlessly integrates the neural network solvers with mechanics governing equations and computational mechanics techniques, offers an excellent alternative for nonlinear solid mechanics problems. MATLAB codes are made available at https://github.com/JinshuaiBai/RPIM_NNS for free downloading.

FEniCS-arclength: A numerical continuation package in FEniCS for nonlinear problems in solid mechanics

FEniCS-arclength: A numerical continuation package in FEniCS for nonlinear problems in solid mechanics

On Solving Nonlinear Elasticity Problems Using a Boundary-Elements-Based Solution Method

The attractiveness of the boundary element method—the reduction in the problem dimension by one—is lost when solving nonlinear solid mechanics problems. The point collocation method applied to strong-form differential equations is appealing because it is easy to implement. The method becomes inaccurate in the presence of traction boundary conditions, which are inevitable in solid mechanics. A judicious combination of the point collocation and the boundary integral formulation of Navier’s equation allows a pure boundary element method to be obtained for the solution of nonlinear elasticity problems. The potential of the approach is investigated in some simple examples considering isotropic and anisotropic material models in the total Lagrangian framework.

A co‐rotational virtual element method for 2D elasticity and plasticity

AbstractThe virtual element method (VEM) allows discretization of the problem domain with polygons in 2D. The polygons can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. VEM applied to linear elasticity problems is now well established. Nonlinear problems involving plasticity and hyperelasticity have also been explored by researchers using VEM. Clearly, techniques for extending the method to nonlinear problems are attractive. In this work, a novel first‐order consistent VEM is applied within a static co‐rotational framework. To the author's knowledge, this has not appeared before in the literature with virtual elements. The formulation allows for large displacements and large rotations in a small strain setting. For some problems avoiding the complexity of finite strains, and alternative stress measures, is warranted. Furthermore, small strain plasticity is easily incorporated. The basic method, VEM specific implementation details for co‐rotation, and representative benchmark problems are illustrated. Consequently, this research demonstrates that the co‐rotational VEM formulation successfully solves certain classes of nonlinear solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

Use of differential forms in the determination of divergence of cross products of tensors in nonlinear solid mechanics

AbstractWe consider the cross product of tensors introduced by R. de Boer (1982) and used by J. Bonet et al. (2016) in describing problems from nonlinear solid mechanics. From the paper by J. Bonet et al., it appears that the cross product of tensors can be used to rearrange the mechanical descriptions and significantly simplify the expressions of selected mechanical quantities encountered in the description of mechanical problems in solid mechanics with large strains. In this paper, we derive some additional properties of the cross product of tensors, which also allow some simplifications of selected mechanical quantities. In particular, we establish a connection between the cross product of operators and differential forms introduced by M. Spivak (1971, 1999). These connections allow further generalization of some expressions. At this point, we take advantage of the important fact that the second differentials of differential forms are zeros. This fact allows further simplification of certain expressions and opens up further new possibilities in the description of mechanical problems in nonlinear solid mechanics.

Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method

The main aim of this work is to present a numerical analysis of convergence and accuracy of the selected generalized perturbation-based schemes in linear and nonlinear problems of solid mechanics. An algorithm for determining the basic probabilistic characteristics has been developed using the iterative generalized stochastic higher-order perturbation method adjacent to symmetrically truncated Gaussian random variables. It has been confirmed that usage of a sufficiently high order of the Truncated Iterative Stochastic Perturbation Technique (TISPT) allows for achieving any desired accuracy in determining up to the fourth-order probabilistic characteristics of static structural response. The semi-analytical probabilistic approach is the reference solution in this study, which is based upon the same composite response functions determined with the use of the Least Squares Method created using specific series of FEM experiments. The entire methodology has been provided for the given extreme value of the coefficient of variation αmax of the input uncertainty source. On the other hand, a selection procedure of the stochastic perturbation method order to achieve 1% numerical accuracy in all up to the fourth-order probabilistic moments has been proposed. Computational experiments include simply supported elastic Euler–Bernoulli beam, a set of steel diagrid structures, nonlinear tension of steel round bar as well as homogenization procedure of some particulate composite.

Advanced Numerical Methods in Computational Solid Mechanics

Efficient numerical solving of nonlinear solid mechanics problems is still a challenging issue which concerns various fields: nonlinear behavior, micromechanics, contact mechanics, damage, crack propagation, rupture, etc [...]

A non‐uniform rational B‐splines enhanced finite element formulation based on the scaled boundary parameterization for the analysis of heterogeneous solids

AbstractThe contribution is concerned with a finite element formulation for the nonlinear analysis of heterogeneous solids in boundary representation. It results in an element with an arbitrary number of curved boundary edges. The curved edges can be parametrized by, for example, non‐uniform rational B‐splines (NURBS). The presented element formulation is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). In contrast to SBFEM, the proposed method uses a numerical approximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of interpolation in scaling direction. Thus, different basis functions can be used for each direction, for example, NURBS basis functions in circumferential and Lagrange basis functions in radial direction. It allows the construction of polygonal elements with an arbitrary number of curved sides, which are described by either whole NURBS curves or NURBS curves' segments. The advantage of the presented element formulation is the flexibility in mesh generation. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.

Isogeometric analysis of 3D solids in boundary representation for problems in nonlinear solid mechanics and structural dynamics

Isogeometric analysis of <scp>3D</scp> solids in boundary representation for problems in nonlinear solid mechanics and structural dynamics

Weakly-invasive LATIN-PGD for solving time-dependent non-linear parametrized problems in solid mechanics

A weakly-invasive version of the LATIN-PGD method is introduced to compute reduced-order models in solid mechanics for time-dependent non-linear parametrized problems under quasi-static assumption. Its main interest is to be easily implemented in any general-purpose industrial finite element software while taking advantage of all facilities offered, including the ability to handle any kind of non-linearities. This way of proceeding provides unified tools – for the construction of reduced-order models in non-linear context – all integrated in one certified product in consistency with the purpose of having end-to-end processes. Possibilities of our approach are illustrated thanks to the first implementation conducted within Simcenter Samcef™ software developed by Siemens Digital Industries Software without ever redeveloping any non-linear part of the software. Finally, interesting performance gains are highlighted on a non-linear time-dependent test-case involving some parameters.

Deep learned one‐iteration nonlinear solver for solid mechanics

AbstractNowadays, there are a lot of iterative algorithms which have been proposed for nonlinear problems of solid mechanics. The existing biggest drawback of iterative algorithms is the requirement of numerous iterations and computation to solve these problems. This can be found clearly when the large or complex problems with thousands or millions of degrees of freedom are solved. To overcome completely this difficulty, the novel one‐iteration nonlinear solver (OINS) using time series prediction and the modified Riks method (M‐R) is proposed in this paper. OINS is established upon the core idea as follows: (1) Firstly, we predict the load factor increment and the displacement vector increment and the convergent solution of the considering load step via the predictive networks which are trained by using the load factor and the displacement vector increments of the previous convergence steps and group method of data handling (GMDH); (2) Thanks to the predicted convergence solution of the load step is very close to or identical with the real one, the prediction phase used in any existing nonlinear solvers is eliminated completely in OINS. Next, the correction phase of the M‐R is adopted and the OINS iteration is started at the predicted convergence point to reach the convergent solution. The training process and the applying process of GMDH are continuously conducted and repeated during the nonlinear analysis in order to predict the convergence point at the beginning of each load step. Through numerical investigations, we prove that OINS is powerful, highly accurate and only needs about one iteration per load step. Thus, OINS significantly saves number of iterations and a huge amount of computation compared with the conventional methods. Especially, OINS not only can detect limit, inflection, and other special points but also can predict exactly various types of instabilities of structures.

Isogeometric analysis of 3D solids in boundary representation for problems in nonlinear solid mechanics and structural dynamics

AbstractThis contribution presents an isogeometric approach in three dimensions for nonlinear problems in solid mechanics and structural dynamics. The proposed approach aligns with the boundary representation modeling technique in CAD. Isogeometric analysis is combined with the parameterization of the scaled boundary finite element method. In this way, the boundary description of 3D solids in CAD can be directly utilized for the analysis in an isogeometric framework. The approximation of the solution on the boundary is based on bi‐variate NURBS. We also approximate the interior of the solid with uni‐variate B‐splines to facilitate the solution of nonlinear problems. The nonlinear case is extended to three dimensions in this present work. The method is further extended to dynamic problems. The mass and damping matrices are derived using the same basis functions. The solution of the entire solid is obtained with the Galerkin method. We study several nonlinear and dynamic problems with simple geometries and arbitrary number of boundaries. Moreover, a fiber reinforced composite serves as demonstration for complex shapes.

A NURBS enhanced polygonal element formulation based on the scaled boundary method for the analysis of solids in boundary representation

AbstractThe contribution is concerned with a numerical element formulation for the nonlinear analysis of solid in boundary representation. Its results in a polygonal element with an arbitrary number of sides. Straight edges are possible as well as curved edges, which are described by e.g. Non‐Uniform Rational B‐Splines (NURBS). The element formulation is based on the so‐called scaled boundary finite element method (SBFEM). The basic idea is to scale the domain's boundary with respect to a scaling center in order to describe the interior domain. In contrast to SBFEM, the proposed method makes use of a numerical aproximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of the interpolation in scaling direction. So, different basis functions can be used for each direction, e.g. NURBS basis functions in circumferential and Lagrange basis functions in radial direction. The advantage of the presented element formulation is the flexibility in mesh generation. The avoidance of so‐called hanging node problems as arbitrary element geometries are possible. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions.

Three dimensional cells with embedded strong discontinuity for material failure analysis by the boundary element method

The Boundary Element Method (BEM) associated with the Continuum Strong Discontinuity Approach (CSDA) can be used as a non-geometric alternative for the modelling of crack growth in physically non-linear three-dimensional problems of solid mechanics. This work makes use of the refereed approach through the implementation of volumetric hexahedral cells with embedded discontinuity that enables the evaluation of discontinuous jumps in the displacement field along a crack discontinuity surface. This is made with an internal equilibrium verification, using a regularized isotropic damage constitutive model. Although this formulation has been extensively studied for two-dimensional problems over the past few years, this is the first time that a three-dimensional extension is presented. The strong discontinuity regime in the cells were imposed directly after the end of the elastic regime, as a typical behaviour of the fracture process of brittle materials. These cells are needed only at the specific region of the solid domain where the cracks propagation is supposed to occur, while the remaining non-discretized regions of domains are considered to work in elastic regime. Some simple numerical examples shows the feasibility of such methodology.

Research on the influence of deformable sheet aluminum alloy rheological behavior on stretch forming process limits

Our research was aimed at modeling a symmetrical wrapping and stable control of the processes of shaping by wrapping a sheet blank along a wrapping punch having a double curvature surface and placed on the table of a virtual stretch-tightening press. The main task of modeling was to determine the deformed state with a uniform change in thickness in different regions of the sheet blank without folds and breaks, but without the influence of the rheological behavior of the deformable aluminum alloy. Here, the LS-DYNA software package is used with the use of an explicit or implicit solver for nonlinear problems of solid mechanics, characteristic of nonequilibrium transitions of the initial dissipative phase due to viscosity and heat transfer. Now it is possible to predict defect formation from a practical point of view, using the forming limit diagram (FLD), which makes it possible with sufficient accuracy to determine the degree of deformation of a given sheet material, at which any defects do not occur. The presented results of the study revealed the features of the influence of the rheological behavior of a wrought sheet aluminum alloy on the limits of the shaping process by a wrap.

A robust and scalable unfitted adaptive finite element framework for nonlinear solid mechanics

In this work, we bridge standard Adaptive Mesh Refinement and coarsening (AMR) on scalable octree background meshes and robust unfitted Finite Element (FE) formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics problems posed on complex geometries, as an alternative to standard body-fitted formulations, unstructured mesh generation and graph partitioning strategies. We pay special attention to those aspects requiring a specialized treatment in the extension of the unfitted h-adaptive Aggregated Finite Element Method (h-AgFEM) on parallel tree-based adaptive meshes, recently developed for linear scalar elliptic problems, to handle nonlinear problems in solid mechanics. In order to accurately and efficiently capture localized phenomena that frequently occur in nonlinear solid mechanics problems, we perform pseudo time-stepping in combination with h-adaptive dynamic mesh refinement and re-balancing driven by a-posteriori error estimators. The method is implemented considering both irreducible and mixed (u/p) formulations and thus it is able to robustly face problems involving incompressible materials. In the numerical experiments, both formulations are used to model the inelastic behavior of a wide range of compressible and incompressible materials. First, a selected set of benchmarks is reproduced as a verification step. Second, a set of experiments is presented with problems involving complex geometries. Among them, we model a cantilever beam problem with spherical hollows distributed in a simple cubic (SC) array. This test involves a discrete domain with up to 11.7M Degrees Of Freedom (DOFs) solved in less than two hours on 3072 cores of a parallel supercomputer.

Intelligent step-length adjustment for adaptive path-following in nonlinear structural mechanics based on group method of data handling neural network

A novel step-length adjustment method for adaptive path-following in geometrically nonlinear problems of solid mechanics is proposed in this paper. The core idea is how to predict the critical points on the equilibrium path properly, and then distribute the points on the path accordingly. We characterize the equilibrium path employing a scalar stiffness parameter. Here, we show how the stiffness parameter could be used to detect the critical points on the path. Then, we employ time-series forecasting based on group method of data handling (GMDH) to predict the stiffness parameter which in fact leads to the prediction of critical points. To adaptively adjust the step-length to the predicted critical point, we present a simple formulation to distribute the points on the equilibrium path. In this paper, a novel adaptive path-following algorithm is presented to trace and predict the path efficiently. The proposed algorithm, minimum residual displacement method (MRDM), arc length method (ALM) and generalized displacement method (GDM) are comparatively investigated for the geometrically nonlinear analysis of structures in both continuum and discrete problems (truss and cylindrical shell). Selected examples present large fluctuations in stiffness. We demonstrate with the numerical examples that the trained neural network can properly predict the critical points on the path. The results also illustrate that our adaptive path-following method is able to distribute converged points properly and trace the equilibrium path with a significantly reduced number of points compared to the analyses with the uniform step-length.

A polyhedral element formulation based on the scaled boundary method for the analysis of nonlinear problems in solid mechanics

AbstractIn this contribution a numerical element formulation for the nonlinear analysis of solids is proposed. The element is defined by a polyhedron with an arbitrary number of polygonal faces. It is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). The SBFEM is a semi‐analytical formulation to analyze problems in linear elasticity. Within this method the fundamental concept is to scale the domain's boundary with respect to a scaling center in order to describe the interior domain. The present element formulation employs the scaling concept twice and in contrast to SBFEM, uses a numerical approximation for the displacement response in both scaling directions. This makes it suitable for the analysis of geometrically and physically nonlinear problems. The advantages of the proposed element formulation is its high flexibility in mesh generation. It avoids the hanging node problems as arbitrary element geometries are possible. For example, using Octree algorithms, a fast and reliable mesh generation can be achieved. In case of curved boundaries the element formulation allows a precise representation of the geometry even with coarse meshes. Some numerical examples are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.

Approximate Solutions to Nonlinear Problems of Solid Mechanics by Quasilinearization Method

The approximate solution to the problem for an infinite plate with the circular hole under creep regime is obtained by the quasilinearization method. Four approximations of the solution for the nonlinear creep problem are found. It is shown that with increasing the number of approximations the solution converges to the limit numerical solution. It is worth to note that the tangential stress reaches its maximum value not at the circular hole but at the internal point of the plate. It is shown that quasilinearization method is the effective method for nonlinear problems. The model problem for steady state creep of a rotating disc is either considered. The high values of the creep exponent are analyzed and it is shown that the high values of the creep exponent require more iterations in the quasilinearization method.

Multiscale analysis of heterogeneous materials in boundary representation

AbstractThis paper deals with a numerical method for modeling heterogeneous materials with nonlinear behavior. Many novel high‐performance materials consist of different components. In the frame of nonlinear behavior the resulting macroscopic material properties cannot be determined without great experimental effort. In order to simulate the nonlinear material behavior, a nested finite element method (FE2) is used. An accompanying homogenization is performed for each load step. The material behavior is determined from the represented volume element (RVE) on the heterogeneous micro structure. The present approach is dealing with a discretization method of the heterogeneous structure. It based on the geometrical data of the interface boundary of the different constituents. A quad‐tree algorithm, in combination with a trimming algorithm is used for the mesh generation. Based on the geometrical data it yields an initial mesh for the analysis. Within the algorithm a staircase approximation of voids or inclusions is circumvented. However, the resulting mesh yields polygonal elements with an arbitrary number of nodes on the element boundary. To this end a novel polygonal element formulation is provided. It is based on the scaled boundary finite element method (SBFEM). The basic idea is to scale the boundary representation in relation to a scaling center. In contrast to SBFEM the present method makes us of an approximation of the displacement response in scaling direction. This enables the analysis of geometrical and material‐related nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of the interpolation in scaling direction. This allows for novel refinement strategies, where e.g. p‐refinement is only applied for the interior element domain. The internal degrees of freedom are eliminated by static condensation. It leads to an element formulation with an arbitrary number of nodes at the element boundary. Thus, the present element is perfectly suited for a quad‐tree meshing algorithm of the RVE. Its hierarchical, tree‐like data structure makes it very useful for adaptive mesh refinements of complex geometries and regions with localized gradients. The classical hanging‐nodes problem for standard elements is omitted by the present element formulation. Some numerical examples show the capability of the formulation with respect to the numerical homogenization of heterogeneous materials.