3D Elasticity Problems Research Articles

Definition of a beam-like reduced order model element by means of a mixed dimensional coupling

The use of Reduced Dimensional Models (RDM) discretized like beams, plates and shell elements drastically decreases the computational cost of solving a full 3D elastic problem with a Finite Element Method (FEM). However, its kinematic assumptions are only applicable to bodies with regular sections or continuous layouts. For the correct analysis of irregular regions, it is necessary to rely on bi-dimensional or solid models that fully reproduce the geometry of the body and its behavior but have a much higher computational cost. The Mixing Dimensional Coupling (MDC) technique allows linking models discretized with elements of different topologies, allowing the possibility of considering the most cost-effective model in each region. This coupling takes place at the interface that delimits both models and relies on the equilibrium of work and reactions on its two faces. In this paper, the formulation is presented for coupling beams with laminar sections and 2D Plane-Stress (PS) models demonstrating its proper behavior. Finally, this coupling is used for defining a new beam element, the Beam-Like Reduced Order Model (BLROM), which is obtained from a Plane-Stress model of their longitudinal section.

Development of quadratic enhanced assumed strain elements for three-dimensional linear elasticity

Enhanced assumed strain (EAS) elements are widely used in the literature to address the issue of locking associated with conventional elements. However, existing literature in the context of three-dimensional (3D) elasticity problems is predominantly restricted to the eight-node linear EAS elements. Thus, existing 3D EAS elements do not exploit the superior performance offered by quadratic elements over linear elements. In the current work, we propose a novel twenty-seven node quadratic EAS element, which, to the best of our knowledge, is the first such attempt in the literature. Additionally, the manuscript also presents a six-node wedge and an eighteen-node wedge EAS element. The proposed elements are derived methodically by investigating the interrelation between the two-field Hellinger–Reissner (HR) and three-field Veubeke–Hu–Washizu (VHW) variational formulations. The robustness and performance of the proposed EAS elements is demonstrated through numerous examples.

Asymptotically correct 3D displacement of the Mooney–Rivlin model using VAM

The current work focuses on the nonlinear analytical analysis (dimensional reduction and recovery relations) of a hyperelastic plate governed by the compressible Mooney–Rivlin model using the Variational Asymptotic Method (VAM). The geometric nonlinearity is accommodated through finite deformations, and generalized 3D warping functions, while material nonlinearity through the hyperelastic material model. VAM mathematically splits the 3D nonlinear elastic problem into the 1D through the thickness analysis and 2D nonlinear plate analysis, using the inherent small parameters. These are the geometric small parameter (ratio of thickness to the characteristic dimension), and the physical small parameter (moderate strains). This work results include the derivation of closed-form analytical expressions of 3D warping functions, 2D nonlinear constitutive relation, and recovery relation to express the 3D displacement field for a plate. The 2D nonlinear constitutive relation is given as an input to the in-house developed 2D nonlinear finite element analysis of the reference surface to determine the 2D displacements and 2D strains. In order to validate the current theory, standard test cases are solved and compared with 3D nonlinear Finite Element Analysis (FEA).

Accuracy verification of fast adaptive galerkin finite volume solver for thermal and mechanical load crack propagation in triangular meshes

This study presents the details of the Galerkin finite volume method for triangular unstructured meshes and its application for the fracture analysis of 2D elasticity problems under simultaneous thermal and mechanical loading. The heat diffusion and elasticity equations are discretized using the Galerkin finite volume method (which, due to omitting matrix manipulation calculations, computations are considerably faster than FEM and XFEM Solvers). The maximum tangential stress criterion is presented after briefly describing the interaction integral formulation used to calculate the stress intensity factors in thermo-mechanical problems. The accuracy of the computed results of the present strategy under thermo-mechanical loads is demonstrated using some benchmark test cases. First, a thermal stress analysis of a plate with an inclined central crack under thermal boundary conditions is performed. Second, the development of a crack under mechanical stress boundary conditions is modeled. Third, crack propagation under both thermal and mechanical boundary conditions is simulated. The present modeling strategy's results are compared with those reported in previous numerical works to verify the accuracy. The stress intensity factors and predicted crack trajectories are utilized to assess the accuracy of computed results and investigate the quality of crack simulation by the proposed numerical method.

A Hybrid Reproducing Kernel Particle Method for Three-Dimensional Elasticity Problems

This study presents a fast meshless method called the hybrid reproducing kernel particle method (HRKPM) for the solution of three-dimensional (3D) elasticity problems. The equilibrium equations of 3D elasticity are divided into three groups of equations, and two equilibrium equations are contained in each group. By coupling the discrete equations for solving two arbitrary groups of equations, the complete solution of 3D elasticity can be obtained. For an arbitrary group of equations, the 3D elasticity problem is transformed into a series of associated two-dimensional (2D) ones, which is solved by the RKPM to derive the discrete formulae. The discrete equations of 2D problems are combined using the difference method in dimension splitting direction. Then, arbitrarily choosing another group of equilibrium equations, the discrete equation of another group of 2D problems can be obtained similarly. By combining the discrete equations for these two groups of 2D problems, the solution to an original 3D problem will be reached. The numerical results show that the HRKPM performs better than RKPM in solution efficiency.

Stochastic virtual element methods for uncertainty propagation of stochastic linear elasticity

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

Extended isogeometric analysis: a two-scale coupling FEM/IGA for 2D elastic fracture problems

Extended isogeometric analysis: a two-scale coupling FEM/IGA for 2D elastic fracture problems

A simple meshfree method based on Trefftz attitude for 2D and 3D elasticity problems

A simple meshfree method is developed for the solution of 2D and 3D elasticity problems in potentially heterogeneous media. The rationale of the method follows that of Trefftz approaches to apply the Partial Differential Equation (PDE) and the boundary conditions in separate steps, while the basis functions can satisfy the PDE. The solution domain is discretized by a regular nodal grid including DOFs as the displacement components. The boundary is also discretized by some boundary points independent from the nodes, making the method applicable for arbitrarily shaped domains without imposing irregularity to the nodal grid. Each node corresponds to a cloud that contains some of its adjacent nodes as well. The overlap of the clouds integrates the displacement and stress components throughout the domain. The governing elasticity PDEs in heterogeneous media have non-constant coefficients, preventing Trefftz techniques to be applicable. The present method, based on equilibrated basis functions, satisfies the PDE in weighted residual approach to extract some bases capable of its approximately satisfaction. The weighting may remove the boundary integrals, so the boundary conditions are simply collocated. The integrals are composed by combination of 1D predefined ones, then removing the numerical quadrature from the solution procedure.

Asymptotic analysis of Timoshenko-like orthotropic beam with elliptical cross-section

This work presents the asymptotically exact analysis of homogeneous-orthotropic prismatic beam with elliptic cross-section. Variational Asymptotic Method (VAM) has been used to systematically reduce the 3D elastic problem of beam analysis into 2D cross-sectional analysis and 1D along the length of beam analysis. This reduction simplifies beam analysis to a great extent. The 2D analysis supplies cross-sectional stiffness constants to the 1D beam problem. This analysis provides closed-form expression of cross-sectional stiffness constants, 3D displacement and strain fields in terms of 1D strain measures. Present work does not make any a priori ad-hoc assumptions about the deformation as it uses VAM and ensures asymptotic correctness of the solution. The results are obtained for Classical and Timoshenko-like beam models loaded with most general loads. These results are validated by comparing them with existing results in the literature and Finite Element Analysis (FEA) results. Current results are in close agreement with FEA results.

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

3D Structural Topology Optimization Using ESO, SESO and SERA: Comparison and an Extension to Flexible Mechanisms

This article investigates the study of Topology Optimization (TO) in 3D elasticity problems to determine the optimal topology by applying the evolutionary methods of Smoothing Evolutionary Structural Optimization (SESO), Sequential Element Rejection and Admission (SERA), and Evolutionary Structural Optimization (ESO). These procedures were implemented in MATLAB code as an extension of Top3d implemented for SIMP by using the eight-node hexahedral finite element formulation in three-dimensional elastostatic structures. The approaches conducted in the present study are demonstrated with numerical examples involving the compliance minimization criterion. Further, a brief synthesis of flexible mechanisms was studied to emphasize the performance of complaint mechanisms measured in terms of two design specifications/functionalities: mechanical and geometrical advantages, which are the highlights of this article. To show the gains of the proposed methods, numerical results obtained are compared with Solid Isotropic Material with Penalization (SIMP) models.

A non-iterative methodology to reconstruct boundary shapes and conditions in isotropic linear elasticity based on the BEM

A non-iterative method is proposed to estimate unknown boundary geometry shapes and boundary conditions in 2D and 3D isotropic linear elasticity problems based on the boundary element method (BEM). A virtual domain with a virtual boundary is introduced to the inverse problem. For the boundary geometry shape identification problem, the displacements of the virtual boundary are obtained through the measurement point displacements and the singular value decomposition is applied to solve the ill-posed matrix. The BEM is used to solve the problem with the displacements of the virtual boundary. The unknown boundary geometry shapes can be estimated by searching the internal point isodisplacement curve. For the boundary condition reconstruction problem, the virtual boundary is overlapped with the real boundary. The displacements of the virtual boundary can also be acquired by the measurement point displacements. The unknown boundary tractions can be calculated by the BEM. Finally, the influences with different virtual domains, measurement points, measurement positions and measurement errors on the results are discussed. The results show that the proposed method is an accurate and efficient way to identify boundary geometry shapes and reconstruct boundary conditions.

Complex-variable, high-precision formulation of the consistent boundary element method for 2D potential and elasticity problems

The collocation boundary element method was recently entirely revisited on the basis of a consistent derivation of Somigliana’s identity in terms of weighted residuals. Both conceptually and for the sake of code implementation, the correct traction force interpolation along generally curved boundaries, as for elasticity problems, leads to the enunciation of an inedited, actually long-sought, convergence theorem as well as to considerable numerical simplifications. Numerical evaluations for two-dimensional problems require exclusively Gauss–Legendre quadrature plus eventual correction terms obtained analytically regardless of the order or shape of the implemented boundary element interpolation. Arbitrarily high precision and accuracy is achievable for low-cost computation, as eventual mesh-subdivision refinements should take place only if the mechanical simulation demands — and not just for numerical evaluations. We now show that considerable simplification is obtained by switching the formulation from real to complex variable. Precision, round-off errors, and accuracy of a given numerical implementation may be kept – identifiably and separately – under control, as assessed for some potential and elasticity examples with extremely challenging topologies. In fact, source-field distances may be arbitrarily small — far smaller than deemed feasible in continuum mechanics, as we resort to nothing else than the problem’s mathematics.

A coupled FETI-BDNM for solving 3D elastic frictional contact problem

Based on the B-differentiable Newton method (BDNM) and finite element tearing and interconnecting (FETI) algorithm, a coupled approach named FETI-BDNM is developed to solve the 3D elastic contact problem with friction. In this method, the contact bodies are decomposed into a number of subdomains. The interfaces between the two adjacent substructures will be skillfully considered as the special contact surfaces that are always in the stick state. By resorting to the generalized inverse of stiffness matrices of the subdomains, the relationships between the relative displacements and forces on contact/interface, i.e., contact flexibility matrices will be computed efficiently. The contact equations, displacement continuity equations and equilibrium equations for each subdomain are expressed in the form of B-differentiable equations (BDEs) uniformly, which can be solved by the damped Newton-Raphson method with 1D Armijo type linear search. The proposed method makes it possible to solve large-scale contact problems without compromising the solution accuracy compared to the original BDNM, and the good convergence of BDEs and parallel scalability of the FETI algorithm will be preserved. Numerical examples validate the accuracy, efficiency, robustness and effects of the number of subdomains on the computational time of the proposed method.

The consistent boundary element method for potential and elasticity: Part III — Topologically challenging numerical assessments for 2D problems

The collocation boundary element method is consistently outlined in a companion (Part I) paper for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. Simple code schemes are proposed in a second companion (Part II) paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high – actually only machine-precision dependent – computational accuracy of all results of interest independently of a problem’s geometry and topology. We present here numerical results and convergence assessments – including numerical illustration of convergence Theorem 1 of the companion paper I – for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source–field distances — maybe with approximations due to a coarse mesh discretization but never introducing unduly singularities. We dare say the present results cannot be replicated by any code implementation other than the ones of our own, which resort to no ad hoc means beyond the problem’s correct mathematics. In fact, we present the objective and controllable tools of separately assessing a mesh discretization for a given practical problem in terms of precision, accuracy and liability to round-off-errors. We also evaluate the distance threshold for a source point to be considered sufficiently close to demand the proposed exact mathematical treatment, as, for large distances, only Gauss–Legendre quadrature is required, be it in terms of quadrature adaptivity of even a fast multipole scheme.

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation.
 Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written.
 The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

The consistent boundary element method for potential and elasticity: Part I — Formulation and convergence theorem

The collocation boundary element method is outlined for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity, as already proposed in the literature. However, this is done in a consistent way that sheds light on some conceptual and implementation-oriented aspects that should be – but are missing – in the textbooks on the subject. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. A rather recreative example of a simple truss element is brought in an Appendix. It enables the demystification of apparently convoluted concepts that might be thought only pertaining to highly elaborated mechanics and mathematics for singularities related to infinity and to ungraspable linear algebra. Simple code schemes based on the outlined concepts are proposed in a companion paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high computational accuracy of all results of interest independently of a problem’s geometry or topology. Numerical results are shown in another companion paper and convergence features are assessed for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source-field distances. Summing up, whether a problem we are attempting to simulate consists of smooth fields or not, we are able to reproduce it — maybe with numerical approximations due to a coarse mesh discretization but never introducing unduly singularities.

Applications of VAM-based equivalent beam model on effective performance of composite sandwich beam with M-shaped folded core

The issue of facesheet-core debonding is expected to be addressed by increasing the bonding platform of the M-shaped folded core. To investigate its effective performance, the three-dimensional traditional equivalent beam (3D-TEB) model and the one-dimensional equivalent beam model (1D-EBM) of the M-shaped folded sandwich beam were established based on the variational asymptotic method. The expression of the original 3D elastic problem was given in variational form. By minimizing the energy functional, the asymptotic solutions of the warping functions and the Euler beam model (zeroth-order approximation) were obtained. Through the static and dynamic analysis of a three-dimensional folded sandwich beam and two equivalent beam models, the effectiveness of the 1D-EBM instead of the original 3D-TEB for static and dynamic analysis was verified. The influences of the location and frequency of the excitation on the dynamic performance of the CSB-MFC were investigated. Compared with 3D-FEM and 3D-TEM, the novelties of cell tailorability and local field recovery within the unit cell are very important features, which significantly facilitates the parametrical analysis of the CSB-MFC. The results showed that the change in the top width of the bonding platform had a particularly significant impact on the torsional stiffness. The wall thickness mainly affected the torsional performance, and the increase in the core height primarily improved the torsional and bending stiffness, resulting in the gradual increase in the fundamental frequency. • The VAM-based 1D-EBM is established to analyze the global response. • The accuracy of the model is verified by static and dynamic examples. • The influence of cell geometric parameters is systematically analyzed. • The dynamic performance of CSB-MFC is investigated based on the equivalent model. • The reduced DOFs of 1D-EBM result in high calculation efficiency.

A new element-free Galerkin method for 3D elasticity problems

A new element-free Galerkin method for 3D elasticity problems

Hybrid TFETI domain decomposition with the clusters joined by faces’ rigid modes for solving huge 3D elastic problems

Hybrid TFETI domain decomposition with the clusters joined by faces’ rigid modes for solving huge 3D elastic problems