Higher order extension of PDS-FEM

Mahendra Kumar Pal; Muneo Hori; Lalith Wijerathne

Abstract

This paper presents the higher order extension of Particle Discretization Scheme (PDS) and its implementation in FEM framework (PDS-FEM)[1, 2, 3] with the aim of simulating brittle cracks in linear elastic solids. PDS-FEM has several attractive features; has a numerically efficient failure treatment, no special treatments required for modeling crack branching, etc. The existing PDS and PDS-FEM are first-order accurate and the objective of this work is to develop a higher order extension while preserving the above mentioned attractive features. A unique property of PDS is the use of conjugate domain tessellations Voronoi and Delaunay to approximate functions and their derivatives, respectively. Let {Φ α } and {Ψ β } be sets of Voronoi and Delaunay tessellation elements of the analyzed domain. Further, let φ α (x) and ψ β (x) be the characteristic functions, and x α and y β be the mother points of elements Φ α and Ψ β , respectively. In higher order PDS, a function is approximated as f (x) ≈ f d (x) = α,n f αn P αn , where {P αn } = {1, x − x α ,. .. , (x − x α) r ,. . .}φ α (x) is a set of polynomial bases with the support of Φ α. The support of P αn 's being confined to the domain of each Φ α , f d (x) consists of discontinuities along Voronoi boundaries. Considering local polynomial approximations over Delaunay tessellation, {Ψ β }, which is the conjugate of Voronoi, PDS defines bounded derivatives for this discontinuous approximation f d (x). Higher order PDS approximates the derivative f, i = ∂f (x) /∂x i as f, i ≈ β,m g βm i Q βm , where {Q βm } = {1, x − x β ,. .. , (x − x β) r ,. . .}ψ β (x). Choosing a suitable sets of polynomial bases for {P αn } and {Q βm }, one can obtain higher order accurate approximations for a given function and its derivatives. The higher order PDS is implemented in the FEM framework to solve the boundary value problem of linear elastic solids. As above explained, PDS approximates a given function and its derivatives as the union of local polynomial expansions defined over the elements of conjugate tessellations. According to the authors' knowledge, this is the first time to use approximations based on local polynomial expansions to solve boundary value problems. With standard benchmark problems in linear elasticity, it is demonstrated that second order accuracy can be obtained by including polynomial bases of first order in {P αn }[3]. Further, exploiting the inherent discontinuities in f d (x), PDS-FEM proposes a numerically efficient treatment for modeling cracks. The efficient crack treatment of higher order PDS-FEM is verified with standard mode-I crack problems. References [1] Hori M, Oguni K, Sakaguchi H, Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of Mechanics and Physics of Solids. 2005; 53-3: 681-703. [2] M.L.L. Wijerathne, Kenji Oguni, Muneo Hori, Numerical analysis of growing crack problem using particle discretization scheme, Int.

Higher order extension of PDS-FEM

M.L.L. Wijerathne ${ }^{1}$ , Mahendra Kumar Pal ${ }^{2}$ , Muneo Hori ${ }^{1}$ and Tsyushi Ichimura ${ }^{1}$
${ }^{1}$ Department of Civil Engineering, The University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656 Japan
${ }^{2}$ Earthquake Research Institute, The University of Tokyo
1-1-1 Yayoi, Bunkyo-ku, Tokyo, 113-0032, Japan

A unique property of PDS is the use of conjugate domain tessellations Voronoi and Delaunay to approximate functions and their derivatives, respectively. Let $\left\{\Phi^{\alpha}\right\}$ and $\left\{\Psi^{\beta}\right\}$ be sets of Voronoi and Delaunay tessellation elements of the analyzed domain. Further, let $\phi^{\alpha}(\boldsymbol{x})$ and $\psi^{\beta}(\boldsymbol{x})$ be the characteristic functions, and $\boldsymbol{x}^{\alpha}$ and $\boldsymbol{y}^{\beta}$ be the mother points of elements $\Phi^{\alpha}$ and $\Psi^{\beta}$ , respectively. In higher order PDS, a function is approximated as $f(\boldsymbol{x}) \approx f^{d}(\boldsymbol{x})=\sum_{\alpha, n} f^{\alpha n} P^{\alpha n}$ , where $\left\{P^{\alpha n}\right\}=\left\{1, \boldsymbol{x}-\boldsymbol{x}^{\alpha}, \ldots,(\boldsymbol{x}-\right.$ $\left.\boldsymbol{x}^{\alpha}\right)^{r}, \ldots\} \phi^{\alpha}(\boldsymbol{x})$ is a set of polynomial bases with the support of $\Phi^{\alpha}$ . The support of $P^{\alpha n}$ 's being confined to the domain of each $\Phi^{\alpha}, f^{d}(\mathbf{x})$ consists of discontinuities along Voronoi boundaries. Considering local polynomial approximations over Delaunay tessellation, $\left\{\Psi^{\beta}\right\}$ , which is the conjugate of Voronoi, PDS defines bounded derivatives for this discontinuous approximation $f^{d}(\mathbf{x})$ . Higher order PDS approximates the derivative $f,_{i}=\partial f(\boldsymbol{x}) / \partial x_{i}$ as $f,_{i} \approx \sum_{\beta, m} g_{i}^{\beta m} Q^{\beta m}$ , where $\left\{Q^{\beta m}\right\}=\left\{1, \boldsymbol{x}-\boldsymbol{x}^{\beta}, \ldots,\left(\boldsymbol{x}-\boldsymbol{x}^{\beta}\right)^{r}, \ldots\right\} \psi^{\beta}(\boldsymbol{x})$ . Choosing a suitable sets of polynomial bases for $\left\{P^{\alpha n}\right\}$ and $\left\{Q^{\beta m}\right\}$ , one can obtain higher order accurate approximations for a given function and its derivatives.

The higher order PDS is implemented in the FEM framework to solve the boundary value problem of linear elastic solids. As above explained, PDS approximates a given function and its derivatives as the union of local polynomial expansions defined over the elements of conjugate tessellations. According to the authors’ knowledge, this is the first time to use approximations based on local polynomial expansions to solve boundary value problems. With standard benchmark problems in linear elasticity, it is demonstrated that second order accuracy can be obtained by including polynomial bases of first order in $\left\{P^{\alpha n}\right\}$ [3]. Further, exploiting the inherent discontinuities in $f^{d}(\mathbf{x})$ , PDS-FEM proposes a numerically efficient treatment for modeling cracks. The efficient crack treatment of higher order PDS-FEM is verified with standard mode-I crack problems.

References

[1] Hori M, Oguni K, Sakaguchi H, Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of Mechanics and Physics of Solids. 2005; 53-3: 681-703.
[2] M.L.L. Wijerathne, Kenji Oguni, Muneo Hori, Numerical analysis of growing crack problem using particle discretization scheme, Int. J. for Numerical Methods in Engineering, Vol.80, pp. 46-73, 2009.
[3] Mahendra Kumar Pal, Lalith Wijerathne, Muneo Hori, Tsyushi Ichimura, Seizo Tanaka,: Implementation of Finite Element Method with higher order Particle Discretization Scheme, J. of Japan Society of Civil Engineers, Ser.A2 ,70(2), pp. 297-305, 2014.

References (3)

Hori M, Oguni K, Sakaguchi H, Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of Mechanics and Physics of Solids. 2005; 53-3: 681-703.

M.L.L. Wijerathne, Kenji Oguni, Muneo Hori, Numerical analysis of growing crack problem using particle discretization scheme, Int. J. for Numerical Methods in Engineering, Vol.80, pp. 46-73, 2009.

Mahendra Kumar Pal, Lalith Wijerathne, Muneo Hori, Tsyushi Ichimura, Seizo Tanaka,: Implemen- tation of Finite Element Method with higher order Particle Discretization Scheme, J. of Japan Society of Civil Engineers, Ser.A2 ,70(2), pp. 297-305, 2014.

Higher order extension of PDS-FEM

Sign up for access to the world's latest research

Abstract

Related papers

Higher order extension of PDS-FEM

References

References (3)

Related papers

Related topics