Solid Mechanics Problems Research Articles

Solving multi-material problems in solid mechanics using physics-informed neural networks based on domain decomposition technology

Physics-informed neural networks (PINNs) are widely used in the field of solid mechanics. Currently, PINNs are mainly used to solve problems involving single homogeneous materials. However, they have limited ability to handle the discontinuities that arise from multi-material, and they lack the capability to rigorously express complex material contact models. We propose a method for solving multi-material problems in solid mechanics using physics-informed neural networks. Inspired by domain decomposition technology, the calculation domain is divided according to the geometric distribution of materials, with different subnetworks applied to represent field variables. This study explains how the invariant momentum balance, kinematic relations, and different constitutive relations controlled by the material properties are incorporated into the subnetworks, and use additional regular terms to describe the contact relations between materials. Various test cases ranging from two-dimensional plane strain problems to three-dimensional stretching problems are solved using the proposed method. We introduce the concept of parameter sharing in multi-task learning (MTL) and incorporate it in the proposed method, which yields additional degrees of freedom in choosing the sharing structure and sharing mode. Compared with common physics-informed neural network algorithms, which are based on fully independent parameters, we develop a network structure with partial sharing structure and all-sharing mode that achieves higher accuracy when solving the example problems.

An assessment of the total Lagrangian material point method: Comparison to conventional MPM, higher order basis, and treatment of near-incompressibility

The total Lagrangian material point method (TLMPM) is a recent modification to the conventional material point method (MPM). It poses the weak form in the undeformed configuration, and has shown great potential in solid mechanics applications because it seamlessly eliminates the so-called cell-crossing instability and overcomes numerical fracture. Despite its promise, it has not been widely adopted by the MPM community yet, and only a handful of research groups have employed it. At the same time, due to TLMPM’s recent appearance, there are open research questions, such as the subject of near-incompressibility that has not been thoroughly studied within its context. In this paper we attempt to strengthen TLMPM’s position in the field by providing an independent assessment of the method in a set of benchmark problems. Furthermore, we evaluate the need of using higher order spline shape functions, which have been shown to be crucial for the success of the conventional MPM. Finally, we present a simple projection technique for addressing solid mechanics problems that fall within the near-incompressible regime. The numerical results show that TLMPM is indeed a strong alternative to conventional MPM, whereas our near-incompressibility approach produces solutions that are free of volumetric locking and pressure oscillations, while introducing very minor modifications to existing codes.

A nonlocal energy‐informed neural network for isotropic elastic solids with cracks under thermomechanical loads

AbstractIn this paper, a nonlocal energy‐informed neural network is proposed to characterize the deformation behaviors of elastic solids with cracks subjected to thermomechanical loads in the framework of ordinary state‐based peridynamics. Based on principle of virtual work, a nonlocal energy approach is developed to recast the solution to peridynamic equilibrium equation as a problem of minimizing the potential energy of the system, which automatically satisfies the traction‐free boundary conditions. Meanwhile, the energy representation of physical system can be treated as the loss function for machine learning methods. Therefore, a nonlocal energy‐informed neural network is constructed to approximate the solution of the system. A distinct advantage of the proposed neural network is that the strain energy is expressed in terms of spatial integration instead of spatial derivatives, which avoids the invalidation of automatic differentiation at the crack surfaces in original physics‐informed neural networks. To demonstrate the convergence and accuracy of the proposed neural network, a series of problems in solid and fracture mechanics are conducted, and compared with results from analytical solutions or classical numerical methods. Additionally, for elastic material containing initial cracks, displacement extrapolation method is encoded into the proposed neural network to evaluate the static stress intensity factor.

INTEGRATED CALCULATION AND EXPERIMENTAL TECHNOLOGY FOR ANALYSIS OF CONTACT INTERACTION WITH ACCOUNT OF ELASTIC INTERMEDIATE LAYER

The work delivers solution to a relevant scientific problem of solid mechanics. It consists in developing theoretical basis for computational and experimental research of contact interaction of complex-shaped bodies with account for micromechanical models of roughness and other intermediate or surface layers. New methods and models have been developed for the analysis of contact interaction of complex-shaped bodies with account for the micromechanical properties of surface and intermediate layers characterized by nonlinear local contact stiffness. The developed methods and models as well as the numerical analysis tools have been applied to a series of model and applied problems. The deformation behaviour of novel materials with network microstructures of one-dimensional elements has been determined. Macroscopical properties of these materials have been evaluated based on the special microscopic models and the homogenization methods. Regularities in contact distribution and its dependence on geometrical and physical factors have been determined for various complexshaped bodies. New design solutions for machine elements that improve their strength and durability have been justified. The developed methods and analysis tools have been introduced into the design of new engineering products. Their implementation resulted in improved technical characteristics of protection, structural and functional elements of transport vehicles of special purpose, gear transmissions, technical equipment, hydrovolumetric drives and so on. The conducted computational and experimental studies showed good agreement with the real response of network materials and the observed behavior of interacting elastic bodies with intermediate contact layer. Keywords: solid mechanics, complex-shaped body, contact interaction, contact pressure, intermediate layer

PyTorch-FEA: Autograd-enabled finite element analysis methods with applications for biomechanical analysis of human aorta

Background and objectivesFinite-element analysis (FEA) is widely used as a standard tool for stress and deformation analysis of solid structures, including human tissues and organs. For instance, FEA can be applied at a patient-specific level to assist in medical diagnosis and treatment planning, such as risk assessment of thoracic aortic aneurysm rupture/dissection. These FEA-based biomechanical assessments often involve both forward and inverse mechanics problems. Current commercial FEA software packages (e.g., Abaqus) and inverse methods exhibit performance issues in either accuracy or speed. MethodsIn this study, we propose and develop a new library of FEA code and methods, named PyTorch-FEA, by taking advantage of autograd, an automatic differentiation mechanism in PyTorch. We develop a class of PyTorch-FEA functionalities to solve forward and inverse problems with improved loss functions, and we demonstrate the capability of PyTorch-FEA in a series of applications related to human aorta biomechanics. In one of the inverse methods, we combine PyTorch-FEA with deep neural networks (DNNs) to further improve performance. ResultsWe applied PyTorch-FEA in four fundamental applications for biomechanical analysis of human aorta. In the forward analysis, PyTorch-FEA achieved a significant reduction in computational time without compromising accuracy compared with Abaqus, a commercial FEA package. Compared to other inverse methods, inverse analysis with PyTorch-FEA achieves better performance in either accuracy or speed, or both if combined with DNNs. ConclusionsWe have presented PyTorch-FEA, a new library of FEA code and methods, representing a new approach to develop FEA methods to forward and inverse problems in solid mechanics. PyTorch-FEA eases the development of new inverse methods and enables a natural integration of FEA and DNNs, which will have numerous potential applications.

The virtual element method

The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape.Nonetheless, the paper is also addressed to readers who have already heard (and possibly read) about VEMs and are interested in gaining more precise information, in particular concerning their application in specific subfields such as ${C}^1$ -approximations of plate bending problems or approximations to problems in solid and fluid mechanics.

Jacobian weighted element differential method for solid mechanics

In this paper, a novel collocation method is proposed to solve the problems in solid mechanics. In the proposed method, the gradients across the element boundaries are evaluated by a weight-averaging algorithm to achieve higher order continuity. Then, the collocation method is employed to discretize the governing equations for all domain nodes, including the nodes at the interfaces of elements for which the earlier collocation methods have to use the traction-equilibrium equations. As a result of this difference, linear and serendipity elements can be used in the proposed method to reduce the total number of freedoms and easily used for very complex geometries. Moreover, the proposed method significantly improves the accuracy compared with the earlier collocation methods. The provided numerical test cases show that the method can also avoid the degeneration of accuracy when odd-order elements are employed, which further demonstrates the advantages of the proposed method. The proposed method are also leads to better convergency in nonlinear cases and iterative matrix solver. By analyzing the coefficient matrix, we found that the proposed method is better conditioned. The proposed method also yields promising advantages in transient problems for explicit time marching because of the advantage of the collocation methods.

Expansion of consistent particle method to solve solid mechanics problems

AbstractOriginally developed for solving fluid dynamics problems, a Lagrangian particle method called the consistent particle method (CPM) is expanded to solid mechanics problems in this article. The motions of particles are governed by the mass conservation and dynamics equations which are solved by the second‐order predictor–corrector time stepping scheme. The spatial derivatives of variables required in solving the governing equations and computing strain increments are obtained by Taylor series expansion. Stress components are updated from strain components according to the material constitutive relation. The physical density of material is directly computed without using particle number density. No additional particles are needed for the simulation of boundaries. For boundary surface that is partially loaded, the inverse distance weighting method is applied to deal with the stress singularity problem due to sudden change of stresses. Numerical examples including shock wave propagation and large deformation of footing penetration are presented to demonstrate the capabilities of CPM in solving solid mechanics problems.

A conservation law consistent updated Lagrangian material point method for dynamic analysis

The Material Point Method (MPM) is well suited to modelling dynamic solid mechanics problems undergoing large deformations with non-linear, history dependent material behaviour. However, the vast majority of existing material point method implementations do not inherit conservation properties (momenta and energy) from their continuum formulations. This paper provides, for the first time, a dynamic updated Lagrangian material point method for elasto-plastic materials undergoing large deformation that guarantees momenta and energy conservation. Sources of energy dissipation during point-to-grid and grid-to-point mappings for FLuid Implicit Particle (FLIP) and Particle In Cell (PIC) approaches are clarified and a novel time-stepping approach is proposed based on an efficient approximation of the Courant-Friedrich-Lewy (CFL) condition. The formulation provided in this paper offers a platform for understanding the energy conservation nature of future/existing features of material point methods, such as contact approaches.

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 [...]

Large surface deformation due to thermo-mechanical effects during cryopreservation by vitrification - mathematical model and experimental validation.

This study presents a simplified thermal-fluids (TF) mathematical model to analyze large surface deformations in cryoprotective agents (CPA) during cryopreservation by vitrification. The CPA deforms during vitrification due to material flow caused by the combined effects of thermal gradients within the domain, thermal contraction due to temperature, and exponential increase in the viscosity of the CPA as it is cooled towards glass transition. While it is well understood that vitrification is associated with thermo-mechanical stress, which might lead to structural damage, those large deformations can lead to stress concentration, further intensifying the probability to structural failure. The results of the TF model are experimentally validated by means of cryomacroscopy on a cuvette containing 7.05M dimethyl sulfoxide (DMSO) as a representative CPA. The TF model presented in this study is a simplified version of a previously presented thermo-mechanics (TM) model, where the TM model is set to solve the coupled heat transfer, fluid mechanics and solid mechanics problems, while the TF model omits further deformations in the solid state. It is demonstrated in this study that the TF model alone is sufficient to capture large-body deformations during vitrification. However, the TF model alone cannot be used to estimate mechanical stresses, which become significant only when the deformation rates become so small that the deformed body practically behaves as an amorphous solid. This study demonstrates the high sensitivity of deformation predictions to variation in material properties, chief among which are the variations of density and viscosity with temperature. Finally, this study includes a discussion on the possibility of turning on and off the TF and TM models in respective parts of the domain, in order to solve the multiphysics problem in a computationally cost-effective manner.

Virtual element formulation for gradient elasticity

The virtual element method has been developed over the last decade and applied to problems in solid mechanics. Different formulations have been used regarding the order of ansatz, stabilization of the method and applied to a wide range of problems including elastic and inelastic materials and fracturing processes. This paper is concerned with formulations of virtual elements for higher gradient elastic theories of solids using the possibility, inherent in virtual element methods, of formulating C1-continuous ansatz functions in a simple and efficient way.

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.

Interpolation-based immersed finite element and isogeometric analysis

We introduce a new paradigm for immersed finite element and isogeometric methods based on interpolating function spaces from an unfitted background mesh into Lagrange finite element spaces defined on a foreground mesh that captures the domain geometry but is otherwise subject to minimal constraints on element quality or connectivity. This is a generalization of the concept of Lagrange extraction from the isogeometric analysis literature and also related to certain variants of the finite cell and material point methods, as well as the concept of free-form deformation from design. Crucially, the interpolation may be approximate without sacrificing high-order convergence rates, which distinguishes the present method from existing high-order finite cell, CutFEM, and immersogeometric approaches. The interpolation paradigm also permits non-invasive reuse of existing finite element software for immersed analysis. We analyze the properties of the interpolation-based immersed paradigm for a model problem and implement it on top of the open-source FEniCS finite element software, to apply it to a variety of problems in fluid, solid, and structural mechanics where we demonstrate high-order accuracy and applicability to practical geometries like trimmed spline patches.

Моделирование напряженно-деформированного состояния кристалла, выращенного методом Бриджмена

The thermal stress in single crystal grown by Bridgman method is studied. The corresponding quasi-static problem of deformable solid mechanics is considered in axisymmetric case. The governing equations are approximated by finite element method. The effect of boundary conditions and material parameters on the stress-strain state of the single crystal is studied numerically.

One Line of Development of the Galerkin Projection Method in Problems of Stationary Solid Mechanics (Review)

One Line of Development of the Galerkin Projection Method in Problems of Stationary Solid Mechanics (Review)

Free element boundary integration method for solving heat conduction and mechanics problems

In this paper, a novel numerical method called the free element boundary integration method (FEBIM) is proposed for solving general engineering problems on the basis of the free element method (FrEM). The proposed method is a weak-form FrEM, which approximates physical quantities through isoparametric elements as in finite element method. The characteristic of FEBIM is that each used isoparametric element can be freely composed by the collocation node and its surrounding nodes. The main difference of FEBIM from FrEM is that the governing equations for internal nodes and the equilibrium equations for boundary points are transformed into boundary integrals in FEBIM. The source point of the local boundary integrals is the collocation node. As a result, FEBIM not only has the advantage of narrower bandwidth over FrEM, but also has higher computational precision, since the second-order derivative is not involved in the process of forming the equations. Numerical examples of heat transfer and solid mechanics problems are given in the paper to verify the correctness and stability of the proposed method.

Finite line method for solving high-order partial differential equations in science and engineering

In this paper, a completely new numerical method, named Finite Line Method (FLM), is proposed for solving general linear and non-linear high-order partial differential equations (PDEs) in science as well as engineering problems in heat conduction and mechanics. In this method, the computational domain is discretized into a number of collocation points as in the free element method (Gao et al., 2019; Liu et al., 2019), and at each collocation point a set of straight or curved lines crossing the point is formed, which is represented by a number of nodes distributed over lines. The shape functions for each line are constructed using the Lagrange interpolation formulation and the first and high order spatial partial derivatives of the shape functions with respect to global coordinates are derived through an ingenious technique. The derived partial derivatives can be directly substituted into the governing differential equations and related boundary conditions to set up the discretized system of equations.FLM is a type of collocation method, not needing any integration to establish the solution scheme. And since the Lagrange interpolation formulation is used to construct the shape functions, arbitrarily high order lines can be easily formulated for solving any types of linear and non-linear high order PDEs and engineering problems. A number of numerical examples for PDEs, heat conduction and solid mechanics problems are given to demonstrate the correctness and stability of the proposed method.

Penalty-based surface-to-surface contact algorithm for SPH method

Contact treatment is a challenging problem for SPH method due to the lack of connectivity between particles. Currently, particle-to-particle contact algorithms are often used to treat contacts in SPH simulation. However, such contact algorithms need to estimate sizes and normal vectors of boundary particles, which may significantly affect computational accuracy and stability. In this paper, a new penalty-based surface-to-surface SPH contact algorithm is proposed. In the algorithm, α-shape method is employed to reconstruct object surfaces. Contacts between objects are detected through box test and crossover test. Contact boundary conditions are imposed by a penalty method. The proposed contact algorithm is applied to calculating a series of solid mechanics problems. Parameter influence on the accuracy of the contact algorithm is analyzed. The results of the proposed contact algorithm are compared with those of particle-to-particle contact algorithm. Numerical results show that the proposed contact algorithm can achieve good accuracy for frictionless and frictional problems and is capable of handling high-velocity impact problems which involve extremely large deformation and material fragmentation, and it shows superior performance than particle-to-particle contact algorithm.