Solid Mechanics Problems Research Articles

Breaking the state of the heart: meshless model for cardiac mechanics.

Cardiac modeling has recently emerged as a promising tool to study pathophysiology mechanisms and to predict treatment outcomes for personalized clinical decision support. Nevertheless, achieving convergence under large deformation and defining a robust meshing for realistic heart geometries remain challenging, especially when maintaining the computational cost reasonable. Smoothed particle hydrodynamics (SPH) appears to be a promising alternative to the finite element method (FEM) since it removes the burden of mesh generation. A point cloud is used where each point (particle) contains all the physical properties that are updated throughout the simulation. SPH was evaluated for solid mechanics applications in the last decade but its capacity to address the challenge of simulating the mechanics of the heart has never been evaluated. In this paper, a total Lagrangian formulation of a corrected SPH was used to solve three solid mechanics problems designed to test important features that a cardiac mechanics solver should have. SPH results, in terms of ventricle displacements and strains, were compared to results obtained with 11 different FEM-based solvers, by using synthetic cardiac data from a benchmark study. In particular, passive dilation and active contraction were simulated in an ellipsoidal left ventricle with the exponential anisotropic constitutive law of Guccione following the direction of fibers. The proposed meshless method is able to reproduce the results of three benchmark problems for cardiac mechanics. Hyperelastic material with fiber orientation and high Poisson ratio allows wall thickening/thinning when large deformation is present.

A Fourier Transform Analysis of Convergence Properties of Multigrid V-Cycle Algorithm

This paper aims to provide an insight on the mechanism for the fast convergence of multigrid (MG) V-cycle algorithms. With this goal, the Fourier transform analysis is implemented to compute quantitatively the frequency spectrum of components of the residual resulted from the MG iterations. MG method has been extensively employed to solve large-scale systems of algebraic equations. Hence, one needs to understand its fast convergence property and scalability mechanism before applying MG method to a system equation, which affects the computational efficiency directly. In a first step, a number of novel features of the MG algorithm are studied, including the relationships among the iteration numbers needed for converged solutions, condition number of the system matrix and the grid-spacing in this paper. In a second step, the complexity of the MG method is researched, and various numerical experiments have been conducted in solving algebraic equations of models in finite element method for both heat transfer problems and solid mechanics problems. It is found that if the number of iterations in different grids is fixed, the optimal iteration mode is determined according to the attenuation of the high-frequency and low-frequency components of the residual. Our results also show that the complexity of the MG method is close to [Formula: see text], meaning that it is a close-to-linear scalable solver, which echoes the findings given by other literature. This study has revealed an important insight that the linear scalability is achieved by mitigating the negative effects (stagnation in convergence) of the large condition number of the system matrix created on the finest grid, using a sequence of coarser grids with smaller and smaller condition numbers.

Evaluation of breaking wave effects in liquid sloshing problems: ANCF/SPH comparative study

This paper is focused on evaluating the effect of breaking waves in liquid sloshing problems. Two fundamentally different approaches, namely the smoothed particle hydrodynamics (SPH) and the finite element (FE) absolute nodal coordinate formulation (ANCF), are used to describe the liquid sloshing response in several sloshing scenarios. The SPH method is a mesh-free numerical technique often used to capture very large displacements in fluid and solid mechanics problems. ANCF finite elements, on the other hand, can be used to develop a non-incremental solution procedure, suited for the nonlinear analysis of flexible bodies undergoing large rotation and large deformation. The fundamental differences between the two approaches and the advantages and limitations of each are discussed. Two benchmark problems, the dam break and sloshing tank, are used to perform a detailed SPH/ANCF quantitative comparative study in different sloshing scenarios. While a good agreement is found between the ANCF and SPH converged solutions for the dam break problem, in the sloshing tank problem the SPH solution underpredicts the amplitude of oscillation of the fluid center of mass and the wave height at a selected probe point. Because one ANCF element can capture complex shapes, nearly 40 times fewer degrees of freedom than the SPH model are needed in both problems. The use of the ANCF models leads to a CPU saving of 70% and 25% in the broken dam and sloshing tank problems, respectively. In the case of the tank problem, the effect of light, moderate, and severe turbulence is examined. The position of the fluid center of mass is computed using the two approaches, and the results obtained are verified using a reference analytical solution. The power spectral density of the center of mass is evaluated using the fast Fourier transform (FFT) to study the effect of breaking waves. The results show that in the case of light and moderate turbulence, the ANCF model allows for accurately averaging the fluid inertia properties, and the obtained solutions are in good agreement with the SPH solutions. In the case of severe turbulence, on the other hand, the mechanical energy dissipation due to fluid mixing and wave breaking, which can only be captured using the SPH method, damps out the sloshing oscillations.

An extended finite volume model for implicit cohesive zone fracture propagation in a poroelastic medium

Models for fluid-driven fractures in a poroelastic medium involve the simultaneous simulation of multiple physical processes—diffusion of pore pressure, poroelastic stresses, fluid injection into a fractured porous medium, and fracture propagation. To simulate the above-mentioned physics, many algorithms and discretization techniques have been used in the past. The finite volume method has been gaining popularity for simulating solid mechanics problems. The discretization algorithms discussed in the past have used a segregated method for solving the poroelastic momentum balance, where iterations are necessary to obtain a converged solution for each component of the displacement vector. This segregated approach, when coupled with fluid-driven fracture propagation, can lead to excessive simulation times because of the coupled nature of the problem.In this study, an implicit formulation was designed to construct a linear system of equation for the components of the displacement vector to solve the poroelastic momentum balance equation (without iterating between components). Within this formulation, a fluid-driven fracture propagation model was implicitly incorporated using a cohesive zone approach. This methodology provided accurate solutions as shown by the validations in this paper where simulated results were compared with various well-known analytical solutions. The implicitness of the new model allowed significant improvements in computation speed when solving multi-physics problems. The simulation speed-ups were found to be 2 to 8 times when compared with the segregated method.

SoftFEM: The Soft Finite Element Method

SummaryWhile the finite element method (FEM) has now reached full maturity both in academy and industry, its use in optimization pipelines remains either computationally intensive or cumbersome. In particular, currently used optimization schemes leveraging FEM still require the choice of dedicated optimization algorithms for a specific design problem, and a “black box” approach to FEM‐based optimization remains elusive. To this end, we propose here an integrated finite element‐soft computing method, ie, the soft FEM (SoftFEM), which integrates a finite element solver within a metaheuristic search wrapper. To illustrate this general method, we focus here on solid mechanics problems. For these problems, SoftFEM is able to optimize geometry changes and mechanistic measures based on geometry constraints and material properties inputs. From the optimization perspective, the use of a fitness function based on finite element calculation imposes a series of challenges. To bypass the limitations in search capabilities of the usual optimization techniques (local search and gradient‐based methods), we propose, instead a hybrid self adaptive search technique, the multiple offspring sampling (MOS), combining two metaheuristics methods: one population‐based differential evolution method and a local search optimizer. The formulation coupling FEM to the optimization wrapper is presented in detail and its flexibility is illustrated with three representative solid mechanics problems. More particularly, we propose here the MOS as the most versatile search algorithm for SoftFEM. A new method for the identification of nonfully determined parameters is also proposed.

A novel face-on-face contact method for nonlinear solid mechanics

The efficient solution of quasistatic contact problems in nonlinear solid mechanics continues to pose some challenges. Traditional node-to-segment methods are not guaranteed to pass the contact patch test, and exhibit non-smooth contact behavior in the presence of sliding. More recent mortar-based methods, which resolve contact interactions over local facet-pairs and typically feature an integral-form gap constraint, result in a more robust treatment of contact in the presence of geometric nonlinearities and large sliding. However, these methods usually require designation of one surface for the interpolation of contact quantities, and are therefore biased. Moreover, reliable iterative solution in the presence of contact constraints remains a challenge for all implicit contact methods. This work presents an unbiased face-on-face contact method using a median-plane methodology and an integral-form gap constraint. The method additionally features a novel subcycle iterative solution strategy that exhibits reliable and efficient convergence performance, even in the presence of large sliding and curved contacting surfaces. Performance of the method is demonstrated through a suite of nonlinear quasi-static contact problems.

Peridynamic Approach to Predict Ductile and Mixed-Mode Failure

The peridynamic theory has been developed to address problems in solid mechanics regarding fracture through its integral non-local basis. It has been successful in predicting brittle cracking, however, uncertainty still remains with regards to mixed mode and ductile fracture. This work presents a study in using peridynamics to simulate fracture in mixed mode or ductile type fractures

A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

The contribution is concerned with a numerical element formulation in boundary representation. It results in a polynomial element description with an arbitrary number of nodes on the boundary. Scaling the boundary description determines the interior domain. The scaling approach is adopted from the so-called scaled boundary finite element method (SBFEM), which is a semi-analytical formulation to analyze problems in linear elasticity. Within this method, the basic idea is to scale the boundary with respect to a scaling center. The boundary, which is denoted as circumferential direction, and the scaling direction span the parameter space. In the present approach, interpolations in scaling direction and circumferential direction are introduced. The interpolation in circumferential direction is independent of the scaling direction. The formulation is suitable to analyze problems in nonlinear solid mechanics. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located at the interior domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows per definition for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other polygonal element formulations, like the virtual element method (VEM). Some benchmark tests show the capability of the element formulation. A comparison to standard and mixed element formulations is presented. The present approach is perfectly suitable to model heterogeneous structures with inclusions and voids. It avoids also staircase approximation of curved boundaries.

A non-local damage approach for the boundary element method

The conventional (local) constitutive modelling of materials exhibiting strain softening behaviour is susceptive to a spurious mesh dependence caused by numerically induced strain localization. Also, for refined meshes, numerical instabilities may be verified, mainly if the simulations are performed by the boundary element method. An alternative to overcome such difficulties is the adoption of the so called non-local constitutive models. In these approaches, some internal variables of the constitutive model in a single point are averaged considering its values of the neighbouring points. In this paper, the implicit formulation of the boundary element method for physically non-linear problems in solid mechanics is used with a non-local isotropic damage model and a very simple averaging scheme, over internal cells, is introduced. It is shown that the analysis become more stable in comparison to the case of a local application of the same model and that the results recover the desired objectiveness to mesh refinement.

A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

AbstractThe contribution is concerned with a numerical element formulation for the analysis of the nonlinear behavior in solid mechanics. It is based on the so‐called scaled boundary finite element method (SB‐FEM). The SB‐FEM is as a semi‐analytical formulation to analyze problems of linear elasticity. The basic idea is to scale the boundary with respect to a scaling center. Hence, the boundary denoted as circumferential direction and the scaling direction span the parameter space. The original SB‐FEM approach is based on an analytical ansatz for the displacement response preserving equilibrium in scaling direction. In the present approach, an interpolation in scaling direction is introduced to account for nonlinear problems. Lagrange interpolations with different polynomial order are therefore discussed. The interpolation in circumferential direction is independent of the scaling direction and is defined by independent Lagrange functions. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located in the interior of the domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows a priori for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other element formulations, like mixed formulations.

B-spline based boundary conditions in the material point method

The material point method is an increasingly popular method for tackling solid mechanics problems involving large deformations. However, there are issues associated with applying boundary conditions in the method and, to date, no general approach for imposing both Neumann and Dirichlet boundary conditions has been proposed. In this paper, a new B-spline based boundary method is developed as a complete methodology for boundary representation, boundary tracking and boundary condition imposition in the standard material point method. The B-spline interpolation technique is employed to form continuous boundaries which are independent of the background mesh. Dirichlet boundary conditions are enforced by combining the B-spline boundaries with an implicit boundary method. Neumann boundary conditions are included by direct integration of surface tractions along the B-spline boundary. This general boundary method not only widens the problems that can be analysed by all variants of the material point method, when implemented using an implicit solver, but is also applicable to other embedded and non-matching mesh approaches. Although the Dirichlet boundary conditions are restricted to implicit methods, boundary representation, tracking and Neumann boundary condition enforcement can be applied to explicit and implicit methods.

An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems

This paper presents a novel element-free smoothed radial point interpolation method (EFS-RPIM) for solving 2D and 3D solid mechanics problems. The idea of the present technique is that field nodes and smoothing cells (SCs) used for smoothing operations are created independently and without using a background grid, which saves tedious mesh generation efforts and makes the pre-process more flexible. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form that requires only discrete values of shape functions that can be created using the RPIM. By varying the amount of nodes and SCs as well as their ratio, the accuracy can be improved and upper bound or lower bound solutions can be obtained by design. The SCs can be of regular or irregular polygons. In this work we tested triangular, quadrangle, n-sided polygon and tetrahedron as examples. Stability condition is examined and some criteria are found to avoid the presence of spurious zero-energy modes. This paper is the first time to create GS-Galerkin weak-form models without using a background mesh that tied with nodes, and hence the EFS-RPIM is a true meshfree approach. The proposed EFS-RPIM is so far the only technique that can offer both upper and lower bound solutions. Numerical results show that the EFS-RPIM gives accurate results and desirable convergence rate when comparing with the standard finite element method (FEM) and the cell-based smoothed FEM (CS-FEM).

A stabilized mixed implicit Material Point Method for non-linear incompressible solid mechanics

In this work a stabilized mixed formulation for the solution of non-linear solid mechanics problems in nearly-incompressible conditions is presented. In order to deal with high material deformation, an implicit Material Point Method is chosen. Such choice allows avoiding the classical limitations of the Finite Element Method, e.g., element tangling and extreme mesh distortion. The proposed mixed formulation, with displacement and pressure as primary variables, is tested through classical benchmarks in solid and geo-mechanics where a Neo-Hookean, a J2 and a Mohr-Coulomb plastic law are employed. Further, the stabilized mixed formulation is compared with a displacement-based formulation to demonstrate how the proposed approach gets better results in terms of accuracy, not only when incompressible materials are simulated, but also in the case of compressible ones.

The enriched modified local green's function method applied to elasto static problems

The Modified Local Green's Function Method (MLGFM) has been applied to solve a large number of solid mechanics problems. It employs the finite element method (FEM) as an auxiliary tool to automatically develop the Green's functions projections (GFp) without knowing the fundamental solutions using conventional Lagrange shape functions to approximate the solution space. Also, improvements have been made in Finite Element Methods aiming to develop approximate solution capabilities of given problems by enriching their solution spaces. A good example of that is the hierarchical finite element approach. In this paper, the Modified Local Green's Function Method is enriched using a hierarchical approach based on Lobatto shape functions. Some standard set of elasto static problems are employed and compared with the MLFGM, classic FEM and analytical solutions.

Shear Modulus of a Fiber Composite with a Transtropic Viscolelastic Matrix and Transtropic Elastic Fiber

When solving the problems of deformation solid mechanics, the inhomogeneous composite material is modeled as homogeneous, with averaged mechanical properties − effective characteristics. The purpose of this paper is to develop a technique for determining the effective shear modulus for a viscoelastic fiber composite with a transtropic matrix and fiber. Their isotropy planes coincide and are perpendicular to the fiber axis. The effective shear modulus is defined as a function of the matrix and fiber mechanical properties and the volume content of each of them in a composite. A unidirectional composite material with a hexagonal fiber stacking scheme and a unit cell consisting of a viscoelastic matrix and elastic fiber is considered. The geometric model of a composite is a combination of two coaxial infinite cylinders: a hollow cylinder, modeling the matrix, and a solid one, modeling the fiber inserted into it. The volume of the hexagonal cell is approximated by the volume of the cylinder. The radius of the cylinder is chosen so that the fiber volume content in the hexagonal cell coincides with the value of this characteristic for the cylindrical cell. To describe the viscoelastic properties of a composite, the ratios of the hereditary Boltzmann-Volterra theory are used. The shear modulus is defined as an integral operator with a difference kernel. Two boundary problems are considered: with regard to the longitudinal shear of a transonic viscoelastic solid cylinder modeling the composite, and the joint longitudinal shear of the hollow and solid cylinders that model the matrix and fiber materials, respectively. It is assumed that the displacements and tangential stresses on the contact surface of the matrix and fiber are continuous. A tangential harmonic load is applied on the outer surface of the cylindrical cell. To solve such problems, the Laplace transform is used. As the matching condition, the equality of displacements on the outer surface of the cylinder is used for the two problems. The application of the proposed technique makes it possible to determine the characteristics of the integral operator describing the shear modulus for a viscoelastic composite material. An instantaneous shear modulus and relaxation core parameters are found as the functions of the known mechanical characteristics of the matrix and fiber. As an example, the characteristics of the shear modulus for a composite material consisting of a rubber matrix and polyamide fiber are determined.

Some a Posteriori Error Bounds for Numerical Solutions of Plate in Bending Problems

For the efficient error control of numerical solutions of the solid mechanics problems, the two requirements are important: an a posteriori error bound has sufficient accuracy and computation of the bound is cheap in respect to the arithmetic work. The first requirement can be formulated in a more specific form of consistency of an a posteriori bound, assuming that it is not improvable in the order and, at least, coincides in the order with the a priori error estimate. Several new a posteriori error bounds are presented, which improve accuracy and reduce the computational cost. Also for the first time a new consistent guaranteed a posteriori error bound is suggested.

Non-singular method of fundamental solutions for elasticity problems in three-dimensions

In this paper, the Non-singular Method of Fundamental Solutions (NMFS) is extended to three-dimensional (3D) isotropic linear elasticity problems. In order to avoid the singularities in the classical Method of Fundamental Solutions (MFS), are the source points outside the problem domain replaced by normalizing the volume integral of the fundamental solutions over the sphere around the singularity on the physical boundary. The derivatives of the fundamental solutions at the singularity, required in the traction boundary conditions, are calculated from three reference solutions of the linearly varying simple displacement fields. The artificial boundary appearing in MFS is with this operations removed in NMFS. A comparison between NMFS and MFS solutions and analytical solutions for two single and two bi-material elasticity problems is used to assess the feasibility and the accuracy of the newly developed 3D method. Although NMFS results are slightly less accurate than MFS results in all spectra of performed tests, all NMFS results converge to the analytical solution. The lack of artificial boundary is particularly advantageous when using NMFS in multibody problems. The developments describe a first use of NMFS for 3D solid mechanics problems.

Phase-field modeling of brittle fracture using an efficient virtual element scheme

This work addresses an efficient low order virtual element method (VEM) for the phase-field modeling of isotropic brittle fracture. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with constitutive terms rooted in fracture mechanics, see Miehe et al. [23] and Miehe et al. [29]. In the presented contribution, we propose a rigorous variational-based framework for the phase-field modeling of brittle fracture in elastic solids undergoing small strains. The key goal here, is the extension towards the recently developed virtual element formulation due to the flexible choice of nodes number in an element which can be changed easily during the simulation process, as outlined in Wriggers et al. [11] and Wriggers et al. [18]. To this end, the potential density is formulated in terms of suitable polynomial functions, instead of computing the unknown shape functions for complicated element geometries, e.g. arbitrary convex or concave polygonal elements. An important aspect of this work is the introduction of an incremental minimization principle, with a novel construction of the stabilization density for the coupled multi-field problem. On the computational side, a robust and efficient monolithic scheme is employed using the software tool AceFEM program in the numerical implementation to compute the unknowns (displacement and crack phase-field), see Korelc and Wriggers [52]. The performance of the formulation is underlined by means of representative examples.

A low order 3D virtual element formulation for finite elasto–plastic deformations

This work addresses an efficient low order 3D virtual element method for elastic–plastic solids undergoing large deformations. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The successful application of the method to non-linear problems such as finite strain elasticity and plasticity in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. This work is concerned with the extensions of the virtual element method to problems of 3D finite strain plasticity. Low-order formulations for problems in three dimensions, with elements being arbitrary shaped polyhedra, are considered. The formulation is based on minimization of a pseudo energy expression, with a generalization of a stabilization techniques, introduced for two dimensional polygons, to the three-dimensional domain. The resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency, accuracy and convergence properties. For comparison purposes, results of the standard finite element method are also demonstrated.

Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures

In recent years, group theory has been gradually adopted for computational problems of solid and structural mechanics. This paper reviews the advances made in the application of group theory in areas such as stability, form-finding, natural vibration and bifurcation of novel prestressed structures. As initial prestress plays an important role in prestressed structures, its contribution to structural stiffness has been considered. General group-theoretic approaches for several problems are presented, where certain stiffness matrices and equilibrium matrices are expressed in symmetry-adapted coordinate system and block-diagonalized neatly. Illustrative examples on structural stability analysis, force-finding analysis, and generalized eigenvalue analysis on cable domes and cable-strut structures are drawn from recent studies by the authors. It shows how group theory, through symmetry spaces for irreducible representations and matrix decompositions, enables remarkable simplifications and reductions in the computational effort to be achieved. More importantly, before any numerical computations are performed, group theory allows valuable and effective insights on the behavior or intrinsic properties of a prestressed structure to be gained.