Improved robustness for nearly-incompressible large deformation meshfree simulations on Delaunay tessellations

Numerical Meth Engineering

Bordas

et al. 2016

It was observed in [1,2] that the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method.In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes.The main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain. This new strain is then used in the computation of the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes to deliver improved accuracy and pass the patch test to machine precision.

“…On writing Equation at the basis functions derivatives level, its right‐hand side can be expressed in terms of the divergence theorem, as follows:

\int_{{normalΩ}_{C}^{h}} ϕ_{a, j} f (boldx) normald V = \int_{{normalΓ}_{C}^{h}} ϕ_{a} f (boldx) n_{j} normald S - \int_{{normalΩ}_{C}^{h}} ϕ_{a} f_{, j} (boldx) normald V .

Section: Stiffness Matrix For Polytopes Using Strain Smoothingmentioning

confidence: 99%

mentioning

confidence: 99%

Linear smoothed polygonal and polyhedral finite elements

Francis

Numerical Meth Engineering

Bordas

et al. 2016

“…To alleviate these integration errors in the VANP method, a smoothing procedure known as quadratically consistent 3-point integration scheme [28] is performed to correct the values of the basis functions derivatives at the integration points. This smoothing procedure was already used in the linear [21] and nonlinear [36] VANP formulations for nearly-incompressible solids. In this paper, we follow the same approach.…”

Section: Numerical Integrationmentioning

confidence: 99%

A volume-averaged nodal projection method for the Reissner–Mindlin plate model

Computer Methods in Applied Mechanics and Engineering

Kobrich

Hale

et al. 2018

Self Cite

We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy basis functions for field variables approximation and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The meshfree approximation is constructed over a set of scattered nodes that are obtained from an integration mesh of three-node triangles on which the meshfree stiffness matrix and nodal force vector are numerically integrated. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed * Corresponding author. Tel: +56 2 2978 4664, Fax: +56 2 2689 6057,

“…Other options for the stiffness matrix in the stability term are possible and will be discussed in Section 4.2. On replacing K by K g E in the second term in (18), the final expression for the stiffness matrix associated with the integration cell is…”

Section: Stiffness Matrixmentioning

confidence: 99%

“…The corresponding second-order accurate integration scheme for four-node tetrahedral meshes is presented in Reference [16]. On adopting the techniques of Duan et al [15,16], Ortiz-Bernardin and coworkers [17,18] presented formulations to treat nearly-incompressible elasticity in the small-deformation and finite-deformation regimes. All these aforementioned integration methods are developed to remove the consistency error from the numerical solution; none of them theoretically guarantee stability.…”

mentioning

confidence: 99%

Consistent and stable meshfree Galerkin methods using the virtual element decomposition

Numerical Meth Engineering

Russo

Sukumar

2017

Self Cite

Over the past two decades, meshfree methods have undergone significant development as a numerical tool to solve partial differential equations (PDEs). In contrast to finite elements, the basis functions in meshfree methods are smooth (nonpolynomial functions), and they do not rely on an underlying mesh structure for their construction. These features render meshfree methods to be particularly appealing for higher-order PDEs and for large deformation simulations of solid continua. However, a deficiency that still persists in meshfree Galerkin methods is the inaccuracies in numerical integration, which affects the consistency and stability of the method. Several previous contributions have tackled the issue of integration errors with an eye on consistency, but without explicitly ensuring stability. In this paper, we draw on the recently proposed virtual element method, to present a formulation that guarantees both the consistency and stability of the approximate bilinear form. We adopt maximum-entropy meshfree basis functions, but other meshfree basis functions can also be used within this framework. Numerical results for several two-dimensional and three-dimensional elliptic (Poisson and linear elastostatic) boundary-value problems that demonstrate the effectiveness of the proposed formulation are presented.