Earthquake research institute

In this work multiscale failure modeling of composites is made using generalized finite element method (GFEM). In this method the global approximation are constructed by combining the local basis with partition of unity functions. The enrichment functions for the GFEM approximation are computed using a proper orthogonal decomposition (POD) technique. The approximation is then used in a two scale Galerkin scheme for failure modeling of composites. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work, a sensitivity analysis of linear elastic cracked structures using two-scale Generalized Finite Element Method (GFEM) is presented. The method is based on computation of material derivatives, mutual potential energies, and direct differentiation. In a computational setting, the discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires the multiplication of the displacement vectors and stiffness sensitivity matrices. By judiciously choosing the velocity field, the method only requires displacement response in a sub-domain close to the crack tip, thus making the method computationally efficient. The method thus requires an exact computation of displacement response in a sub-domain close to the crack tip. To this end, in this study we have used a two-scale GFEM for sensitivity analysis. GFEM is based on the enrichment of the classical finite element approximation. These enrichment functions incorporate the discontinuity response in the domain. Three numerical examples which comprise mode-I and mixed mode deformations are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method.

This paper studies the extension of particle discretization scheme (PDS) in order to improve finite element method implemented with this discretization scheme (PDS-FEM). Polynomials are included in the basis functions, while original PDS uses a characteristic function or zero-th order polynomial only. It is shown that including 1st order polynomials in PDS, the rate of the convergence reaches the value of 2 even for the derivative. 1st order polynomials are successfully included in PDS-FEM. A numerical experiment is carried out by applying 1st order PDS-FEM, and the improvement of the accuracy is discussed.

This paper presents the higher order extension of Particle Discretization Scheme (PDS) and its implementation in FEM framework (PDS-FEM) to  solve boundary value problems of linear elastic solids, including brittle cracks. Higher order PDS defines an approximation $f^d(\mathbf{x})$ of a function $f(\mathbf{x})$, defined over domain $\Omega$, as the union of local polynomial approximation of $f(\mathbf{x})$ over each Voronoi tessellation elements of $\Omega$.  The support of the local polynomial bases being confined to the domain of each Voronoi element, $f^d(\mathbf{x})$ consists of discontinuities along each Voronoi boundaries. Considering local polynomial approximations over elements of Delaunay tessellation, PDS define bounded derivatives for this discontinuous $f^d(\mathbf{x})$. Utilizing the inherent discontinuities in $f^d(\mathbf{x})$, PDS-FEM proposes a numerically efficient treatment for modeling cracks. This novel use of local polynomial approximations in FEM is verified with a set of linear elastic problems, including mode-I crack tip stress field.

This paper presents implementation of higher order PDS (HO-PDS) in FEM framework (HO-PDS-FEM) to solve a boundary value problems involving cracks in linear elastic bodies. Further, an alternative approach based on curl free restriction to extend the current PDS is also presented. This alternative curl-free implementation is scrutinized and compared with a former proposal for HO-PDS whose derivative is not guarantee to satisfy curl free condition. Analysis of traditional plate with a hole problem shows that curl free implementation does not have any specific advantage. Further, techniques for modeling cracks in HO-PDS-FEM are presented. Comparison of two formulations with mode-I crack problem indicates that former proposed HO-PDS-FEM is superior to the proposed curl free formulation, and there is a significant improvement compared to 0 th-order PDS-FEM.

We present the development a higher order extension of Particle Discretization Scheme (PDS) and its implementation in FEM (PDS-FEM) for modeling 2D and 3D brittle elastic cracks. Numerically efficient treatment to model discontinuities is a salient feature of originally proposed PDS-FEM [1]. The higher order extension [2] brings two more attractive features; higher rate of convergence and increased accuracy of stress state in the vicinity of traction free crack surfaces. A peculiar property of PDS is the usage of conjugate domain tessellation schemes to approximate a given function and its derivatives; we use Voronoi and Delaunay tessellation pair for function and derivatives, respectively. A given function f (x x x) is approximated as f (x x x) ≈ f d (x x x) = ∑ α,n f α P αn (x x x)φ(x x x), where Φ α denotes the characteristic function of α th Voronoi element and P αn (x x x) denotes its n th polynomial base. Similarly, derivatives are approximated with a suitable set of base polynomials and characteristic functions, Ψ β 's, of Delaunay elements. The use of characteristic functions Φ α confines the support of the polynomial bases to the support of each tessellation element, making f d (x x x) discontinuous and giving rise to PDS's particle nature. Though f d (x x x) is discontinuous, PDS defines bounded approximations for derivatives, by approximating those with Delaunay tessellation which is the conjugate of Voronoi. The numerous discontinuities in f d (x) along Voronoi boundaries are utilized to efficiently model discontinuities when solving the boundary value problems. As an example, all it requires to introduce a crack along a Voronoi boundary is recalculating element stiffness matrix of relevant Delaunay element, ignoring the contribution from an infinitesimally thin neighbourhood of the Voronoi boundary to be broken. This surely is much lighter than competing methods in FEM. The proposed higher order extension is verified with several standard benchmark problems involving elastic deformation and mode-I cracks in 2D and 3D. While the benchmark problems without cracks produced the expected ideal convergence rate, those with mode-I cracks produced nearly ideal results. For the mode-I crack problems, higher order PDS-FEM with first order polynomial bases for approximating displacements produced the rate of convergence of 1.7 for J-integral; the ideally expected rate is 2. Further, it is demonstrated that the proposed higher order extension significantly improve the state of stress in the vicinity of traction free crack surfaces, compared to that of former 0 th order PDS-FEM[1]. References [1] M.L.L. Wijerathne, Kenji Oguni, Muneo Hori, Numerical analysis of growing crack problem using particle discretization scheme, Int.

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.