Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative
Introduction
In the natural world as well as in the engineering fields, various kinds of discontinuities occur over an interface. They are well known to be difficult to analyze accurately and robustly in numerical computations when the interface is arbitrarily placed on a computational domain. In developing a method to treat such arbitrary discontinuities over interfaces, it is important to construct the approximation with a derivative discontinuity and to design a robust scheme [4], [12], [13], [21], [23], [25].
In recent decades, for problems involving discontinuities on a curve in 2-D or a surface in 3-D, the immersed boundary method (IBM) by Peskin [25] and his coworkers has been developed and applied to many biological problems. In this case, the discontinuity on a fluid–solid interface yields the Dirac delta interaction force and IBM provides a means to effectively distribute the concentrated force to the vicinity of the interface. Using a diffuse numerical Dirac delta function in IBM takes advantage of lower cost, while its trade-off is the loss of accuracy. To provide better accuracy in treating discontinuities across the interface, the immersed interface method (IIM) was proposed by LeVeque and Li [13], [14]. Additionally, attempts to extensions of IIM have been made for elliptic equations with variable coefficients as seen in [3], [26]. For the same purpose of capturing sharp discontinuities, the boundary condition capturing method in [19] that stems from the ghost fluid method (GFM) by Fedkiw et al. [5] has been applied to the variable coefficient Poisson equation with Dirichlet boundary conditions on the irregular interface [6]. A few years ago, the convergence of the GFM for elliptic equations with interfaces was proved in [20] by using a weak formulation.
However, we still need the pseudo-spectral approximation for the interfacial discontinuities to calculate the approximated sharp discontinuous values on both sides of the interface. The finite element approximation can be one candidate in this viewpoint. The immersed finite element method (IFEM) by Liu et al. [15], [27] is a method to model the interfacial force on a body induced by the background flow where an Eulerian mesh is used for the flow and Lagrangian mesh for the moving body. In IFEM, both fluid and solid domains are modeled with the finite element method and the continuity between fluid and solid subdomains is enforced via the interpolation of the velocities and the distribution of the interaction forces with the reproducing Kernel particle method (RKPM) delta function [16], [17]. In the extended finite element method (XFEM) [24] to analyze the moving discontinuities in function and gradients, the jump function and crack tip singularities are introduced to approximate the discontinuous displacements for a crack. An important theoretical result in finite element methods for the elliptic problems with discontinuous coefficients has been reported by Babus˘ka [1]. According to his early work, sub-optimal convergence rate of the numerical solution in the H1 norm is of when the interface runs through the interior of elements. This means that no treatment of discontinuity can guarantee optimal convergence. In [8], a discontinuity treatment is developed for the variable coefficient elliptic equation even with non-smooth interfaces and the convergences of various examples were tested numerically. As another example of a pseudo-spectral discontinuity treatment, the discontinuous reproducing kernel element method can be seen in [22].
In this paper, keeping the meshfree point collocation method in mind, we first develop meshfree pseudo-spectral approximations of continuous functions with a finite jump of the normal derivative across the interface (i.e. a weak discontinuity). Next, using these approximations, we use the point collocation schemes to solve elliptic equations with discontinuous coefficients and/or the singular layer sources over an interface.
Based on the exact extraction of the leading singular behavior from the function having the interfacial discontinuities, we apply the approximation to the meshfree point collocation method to obtain the numerical solution of elliptic problems with discontinuous coefficients and/or singular layer sources over an interface. The important issue in this approach is to make a discretization of the interface condition. Our approximations are defined at any point in the problem domain, so that the modeling nodes for the interface can be placed arbitrarily.
On the other hand, since the interface conditions are defined on the interface, the linear interface element approximation is introduced only for the approximation of the unknown jump in the normal derivative. According to the numerical results obtained, the proposed approximation yields a sharp approximation of derivatives of the numerical solution without smearing near the interface. Moreover, it is so robust that the optimal convergence for the extrinsic meshfree derivative of the numerical solution can be maintained independent of the interface discretization for the unknown jump approximation and the jump values of the discontinuous coefficients across the interface in the partial differential equations. Although the numerical experiments were done in 2-D problems, the method proposed in this paper is expected to be directly applicable to 3-D once the interfacial surface is modeled effectively enough to approximate a function on it.
Section snippets
Discontinuous meshfree approximation
Let Ω be a bounded domain in and be an admissible set of nodes in where d is the space dimension; ρx in this paper designates the dilation function which takes the place of the dilation parameter and depends on the set Λ of nodes: it has to be a continuous and positive function on compatible with the completeness of the basis polynomials up to order m. By the admissible set of nodes in the above, we mean that there exists a dilation function ρx such that the moment
Point collocation schemes using the extrinsic meshfree approximation
In most cases, the normal derivative jumps across the interface are not known a priori. When the normal derivative jumps are given in advance, the approximation formula (24) can be easily implemented in the meshfree point collocation method without adding degrees of freedom. In contrast to this case, if the normal derivative jump is unknown, then how to discretize the function in (24) must be considered. Although this can be accomplished through modeling of the interface, it is only used
Conclusions
New approximations for the interfacial derivative discontinuities across the interface for meshfree methods are proposed. Using the point collocation strategy based on these new approximations, elliptic partial differential equations with discontinuous coefficients are solved effectively. The accuracy of the extrinsic meshfree approximations is shown through numerical examples. The point collocation scheme combined with the extrinsic meshfree approximations provides robust numerical results for
Acknowledgement
This work was supported by the National Science Foundation and Army Research Office under Grant. The support of the KRF under the grant of KRF–2005–015–C00052 is gratefully acknowledged.
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Visiting Scholar in 2004, Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA.