Adaptive quadtree polygonal based edge-based smoothed finite element method for quasi-incompressible hyperelastic solids (original) (raw)
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Linear smoothed finite element method for quasi-incompressible hyperelastic media
International Journal of Advances in Engineering Sciences and Applied Mathematics, 2021
This work presents a linear smoothing scheme over high-order triangular elements within the framework of the cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme is that it unlike the classical SFEM, it does not require the subdivision of the finite element cells into smoothing sub-cell. The other features of the classical SFEM are retained, such as: it does not require an explicit form of the derivatives of the basis functions, all the computations are done in the physical space, and the results are less sensitive to mesh distortion. A series of benchmark tests are done to demonstrate the validity and the stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using quadratic triangular element and the exact solutions.
An adaptive hybrid stress transition quadrilateral finite element method for linear elasticity
In this paper, we discuss an adaptive hybrid stress finite element method on quadrilateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we derive a priori error estimation for the 5-node transition hybrid stress element to show that it is free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the Lame constant lambda\lambdalambda. We introduce, for quadrilateral meshes, refinement/coarsening algorithms, which do not require storing the refinement tree explicitly, and give an adaptive algorithm. Finally we provide some numerical results.
Strain smoothing for compressible and nearly-incompressible finite elasticity
Computers & Structures, 2017
We present a robust and efficient form of the smoothed finite element method (S-FEM) to simulate hyperelastic bodies with compressible and nearly-incompressible neo-Hookean behaviour. The resulting method is stable, free from volumetric locking and robust on highly distorted meshes. To ensure inf-sup stability of our method we add a cubic bubble function to each element. The weak form for the smoothed hyperelastic problem is derived analogously to that of smoothed linear elastic problem. Smoothed strains and smoothed deformation gradients are evaluated on sub-domains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Numerical examples are shown that demonstrate the efficiency and reliability of the proposed approach in the nearly-incompressible limit and on highly distorted meshes. We conclude that, strain smoothing is at least as accurate and stable, as the MINI element, for an equivalent problem size.
By using the strain smoothing technique proposed by Chen et al. [1] for meshless methods in the context of the finite element method (FEM), Liu et al. [2] developed the Smoothed FEM (SFEM). Although the SFEM is not yet well-understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically the convergence properties in which conditions strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial [cases (a) and (b)], but that non-polynomial enrichment of type (c) lead to inferior methods compared to standard enriched FEM (e.g. XFEM).
Adaptive Poly-FEM for the analysis of plane elasticity problems
International Journal for Computational Methods in Engineering Science and Mechanics, 2017
In this work we present polygonal finite element method (Poly-FEM) for the analysis of two dimensional plane elasticity problems. The generation of meshes consisting of nā sided polygonal finite elements is based on the generation of a centroidal Voronoi tessellation (CVT). An unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set is generated whereby the method also includes tessellation of nonconvex domains.In this work, a patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses is proposed for obtaining the smoothed finite element stresses. A recovery type a ā posteriori error estimator that estimates the energy norm of the error from the recovered solution is then adopted for the polygonal finite element method. The refinement of the polygonal elements is then made on an region by region basis through a refinement index. For the numerical integration of the Galerkin weak form over polygonal finite element domains we resort to classical Gaussian quadrature applied to triangular sub domains of each polygonal element.
Adaptive quadtree edge-based smoothed finite element method for limit state analysis of structures
International Journal of Mechanics and Materials in Design, 2024
This study presents an efficient numerical approach for pseudo-lower bound limit analysis of structures. The total stress field is decomposed into two components: an elastic component associated with the safety factor and a self-equilibrating residual component. Subsequently, equilibrium conditions within the optimization problem are satisfied in a weak manner. The application of the adaptive quadtree edge-based smoothed finite element method (ES-FEM), combined with the transformation into the second-order cone programming (SOCP) form, ensures the resulting optimization problem remains minimal in size. Moreover, employing a yield stress-based adaptive strategy in the proposed procedure either accurately provides limit loads with low computational effort or effectively predicts the collapse mechanism through the concentration of elements after mesh refinement progress. The investigation of a series of numerical tests confirms the effectiveness and reliability of the proposed method.
Computational Mechanics, 2009
An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses using the von-Mises yield function and the Prandtl-Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of the FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM.
International Journal for Numerical Methods in Engineering, 2012
We present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions. This finite element formulation is based on the Eulerian description of motion and the transport of the deformation gradient. When modeling a nearly incompressible solid, the transport of the deformation gradient is decomposed into its isochoric part and the Jacobian determinant as independent fields. A homogeneous isotropic hyperelastic solid is assumed and B-splines-based finite elements are used for the spatial discretization. A variational multiscale residual-based approach is employed to stabilize the transport equations. The performance of the scheme is explored for both compressible and nearly incompressible applications. The numerical results are in good agreement with theory illustrating the viability of the computational scheme.
International Journal for Numerical Methods in Engineering, 1990
A modified global approach to choosing stress terms for hybrid finite elements in plane stress problems is based on the known requirement of minimum number of stress parameters. Let nS be the number of independent B-stress parameters, n, the number of nodal displacements and nR the number of rigid body degrees of freedom, then the satisfaction of the criterion na 2 n, -nR of the assembled structure instead of the individual element enables the reduction of np. New rectangular hybrid transition elements applied in adaptive mesh refinement and a new eight node rectangular invariant element including only 12 / I parameters, based on the modified criterion, are presented here.
A Smoothed Finite Element Method for Mechanics Problems
Computational Mechanics, 2006
In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than 180 ā¢ and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.