Spline-based meshfree method with extended basis

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Highlights

  • Extension is performed to stabilize the basis of spline-based meshfree method.

  • Nitsche's method was employed to impose essential boundary conditions.

  • The validity of the proposed method was proved by error analysis.

Abstract

In this work, an extension has been performed on the analysis basis of spline-based meshfree method (SBMFM) to stabilize its solution. The potential weakness of the SBMFM is its numerical instability from using regular grid background mesh. That is, if an extremely small trimmed element is produced by the trimming curves that represent boundaries of the analysis domain, it can induce an excessively large condition number in global system matrix. To resolve the instability problem, the extension technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche's method is implemented. Using numerical examples, the presented SBMFM with extended basis is shown to be valid and effective. Moreover, the condition number of the system is well-managed guaranteeing the stability of the numerical analysis.

Introduction

Many researchers have exercised a great deal of effort towards reducing the computational costs of creating analysis models from Computer Aided Design (CAD) models. Some researchers focused on achieving an efficient framework for computational simulation by unifying the basis functions involved through the simulation. Kagan et al. proposed an integrated mechanically-based CAE system (Kagan et al., 1998, Kagan et al., 2003, Kagan and Fischer, 2000). Natekar et al. (2004) introduced constructive solid analysis based on the Constructive Solid Geometry (CSG) technique. They employed the partition of unity-based approximation to deal with the overlapping regions. Recently, Hughes et al. (2005), Cottrell et al. (2009) proposed an isogeometric analysis framework that integrates CAD and Finite Element (FE) analysis. So far, Non-Uniform Rational B-spline (NURBS) is mainly employed in isogeometric analysis since NURBS has been widely used as the basis function in CAD (Hughes et al., 2005, Cottrell et al., 2009). Using the same basis function of NURBS in FE analysis, the mesh generated in the CAD system can be used directly, and geometries are exactly preserved throughout the analysis. Moreover, the NURBS basis function has a controllable high continuity (higher than C1) across element boundaries (Cottrell et al., 2007), and is more robust regarding mesh distortion (Lipton et al., 2010). Due to the various advantages that NURBS provides, isogeometric analysis is implemented in various fields of computational mechanics, including structural vibration (Cottrell et al., 2006), fluid–structure interaction (Bazilevs et al., 2008), shape and topology optimization (Seo et al., 2010a, Seo et al., 2010b, Ha et al., 2010, Wall et al., 2008), shell structure analysis (Uhm and Youn, 2009, Kiendl et al., 2009, Nguyen-Thanh et al., 2011, Benson et al., 2010), and contacts (Temizer et al., 2011, Temizer et al., 2012, Kim and Youn, 2012, Lu, 2011).

For the seamless integration of CAD systems and NURBS-based isogeometric analysis, Kim et al. introduced trimmed surface analysis (Kim et al., 2009, Kim et al., 2010). The trimmed surface analysis in an isogeometric analysis framework, directly utilized trimming information imported from the CAD system, and has been successfully applied to problems involving complex topologies. Exploiting the concept of trimmed surface analysis, Kim and Youn introduced the spline-based meshfree method (SBMFM) (H.-J. Kim and S.-K. Youn, 2012). The point of this method was to build an analysis domain by trimming operations so that the boundaries are represented by trimming curves only in a single NURBS patch (in two-dimensional cases). By doing so, the NURBS patch is employed as a background mesh. Unlike conventional meshfree methods, the SBMFM has no ambiguity for constructing basis functions and for describing domain boundaries. The basis function for analysis is readily constructed using the NURBS background mesh. Above all, domain boundaries are easily constructed using the trimming operations. The SBMFM has been applied to geometrically nonlinear problems and demonstrated its effectiveness.

There is a numerical method that similarly employs a spline background mesh: weighted extended B-splines (WEB-splines) proposed by Höllig et al., 2001, Höllig et al., 2005, Höllig and Reif (2003), Höllig (2003). The good characteristic of the WEB-splines is its well-conditioned system matrix. In Fig. 1, the box indicates one of the elements with a very small area. This element can cause numerical instability due to the large condition number of the global system matrix. That is, a relatively small contribution to the global system matrix amplifies the condition number of the system. This is a chronic problem of methods with a structured background mesh, such as immersed (or embedded) boundary method (Peskin, 1972, Peskin, 2002), fictitious domain method (Glowinski et al., 1994) and immersed B-spline techniques (Embar et al., 2010, Schillinger and Rank, 2011, Sanches et al., 2011, Rank et al., 2012, Rüberg and Cirak, 2012); that have becoming popular recently in conjunction with isogeometric analysis. Generally, a large condition number is to be avoided since this induces a highly sensitive numerical solution. Also, when iterative solvers are used, a numerical solution can be inaccurate showing a slow convergence. To resolve this problem, in the WEB-splines, the basis function whose contribution is relatively small is merged with neighboring basis functions, i.e., the neighboring basis functions are extended. As a consequence, a solution field is represented by extended basis functions whose contribution is not so small, and the condition number of the global system matrix is stabilized; preserving an optimal approximation order when smooth weight function is used (Höllig et al., 2001, Höllig et al., 2005, Höllig and Reif, 2003, Höllig, 2003).

In the SBMFM, it is highly probable that trimmed elements with very small areas can exist as WEB-splines. Thus, if we can extend the basis function in the SBMFM, the analysis will be more robust numerically. In this work, the extension of WEB-splines was adopted to stabilize the SBMFM. The proposed approach will be named ‘extended SBMFM’ which is SBMFM with extended analysis basis.

The difference between the WEB-splines and extended SBMFM is that extended SBMFM does not employ weight functions to describe the analysis domain boundary; whereas the extended SBMFM employs trimming information imported from the CAD system to describe the analysis domain boundary. As a consequence, a different scheme for imposing essential boundary conditions should be considered for the extended SBMFM, where basis functions do not vanish outside the analysis domain due to a non-weighted spline space of the extended SBMFM. In this work, Nitsche's method is employed and it will be discussed in Section 3.2.

The remainder of this paper is organized as follows. In Section 2, the basics of B-splines and NURBS will be briefly reviewed and trimmed surface analysis will be introduced. In Section 3, the extension process will be introduced and the essential boundary condition imposition scheme, which is appropriate for extended SBMFM, will be discussed. In Section 4, the validity of the extended SBMFM will be illustrated using verification examples. Finally, conclusions and future work will be discussed in Section 5.

Section snippets

B-splines and NURBS

In this work, a recurrence formula will be employed to construct the B-spline basis functions (Piegl and Tiller, 1997). A non-decreasing sequence of real numbers U is the only information we need in order to construct B-spline basis functions.U={u0un+p}. In Eq. (1), U is the knot vector, and each component of U is called a knot. The number of control points is n; while p is the degree of B-spline basis functions. If the intervals between knots are equal, the knot vector is labeled uniform. If

Formulation of the extension process

In this section, the formulation of SBMFM with extended basis will be described by employing the extension concept of non-uniform WEB-splines (Höllig and Reif, 2003). Throughout the formulation, the following Poisson problem will be considered.Δu=fin Ωu=udon Γdun=gnon Γn, where Ω is the analysis domain, Γd the domain boundary with the Dirichlet boundary condition and Γn the domain boundary with the Neumann boundary condition. The solution for the problem in Eq. (7) is approximated as Eq. (8).

Verification examples

The validity of the present method will be examined with verification examples. The convergence behavior will be tested using a numerical example with an exact solution. The present method also will be applied to an example with a little bit complicated topology imported from the CAD system, and the condition number of the stiffness matrix will be compared with that of conventional SBMFM.

Conclusions and future work

In this work, the extended SBMFM is presented. We focused on the potential instability of conventional SBMFM due to extremely small trimmed basis functions. To get rid of the instability, the extension of the basis function was performed. Through the extension process, the basis functions with small remaining support were merged with neighboring basis functions, so that the instability could be eliminated. In order to impose essential boundary conditions, Nitsche's method was employed.

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0015469).

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    This paper has been recommended for acceptance by Thomas Sederberg.

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