On the enhancement of computation and exploration of discretization approaches for meshless shape design sensitivity analysis

A new implementation of Reproducing Kernel Particle Method (RKPM) is proposed to enhance the process of shape design sensitivity analysis (DSA). The acceleration process is accomplished by expressing RKPM shape functions and their derivatives explicitly in terms of kernel function moments. In addition, two different discretization approaches are explored elaborately, which emanate from discretizing design sensitivity equation using the direct differentiation method. Comparison of these two approaches is made, and the equivalence of these two superficially different approaches is demonstrated through two elastostatics problems. The effectiveness of the enhanced RKPM is also verified by comparison of consumption of computer time between the classical method and the improved method.     

A unified approach to the mathematical analysis of generalized RKPM,gradient RKPM,and GMLS

It is well-known that the conventional reproducing kernel particle method (RKPM) is unfavorable when dealing with the derivative type essential boundary conditions [1], [2], [3]. To remedy this issue a group of meshless methods in which the derivatives of a function can be incorporated in the formulation of the corresponding interpolation operator will be discussed. Formulation of generalized moving least squares (GMLS) on a domain and GMLS on a finite set of points will be presented. The generalized RKPM will be introduced as the discretized form of GMLS on a domain. Another method that helps to deal with derivative type essential boundary conditions is the gradient RKPM which incorporates the first gradients of the function in the reproducing equation. In present work the formulation of gradient RKPM will be derived in a more general framework. Some important properties of the shape functions for the group of methods under consideration are discussed. Moreover error estimates for the corresponding interpolants are derived. By generalizing the concept of corrected collocation method, it will be seen that in the case of employing each of the proposed methods to a BVP, not only the essential boundary conditions involving the function, but also the essential boundary conditions which involve the derivatives could be satisfied exactly at particles which are located on the boundary.     

Shape optimization using reproducing kernel particle method and an enriched genetic algorithm

Combining Reproducing Kernel Particle Method (RKPM) with the proposed Multi-Family Genetic Algorithm (MFGA), a novel approach to continuum-based shape optimization problems is brought forward in this paper. Taking full advantage of the features of meshfree method and the merits of MFGA, the new method solves shape optimization problems in such a unique way that remeshing is avoided and particularly the computation burden and errors caused by sensitivity analysis are eliminated completely. The effectiveness, versatility and performance of the proposed approach are demonstrated via three 2-D numerical examples.     

A remedy to gradient type constraint dilemma encountered in RKPM

A major disadvantage of conventional meshless methods as compared to finite element method (FEM) is their weak performance in dealing with constraints. To overcome this difficulty, the penalty and Lagrange multiplier methods have been proposed in the literature. In the penalty method, constraints cannot be enforced exactly. On the other hand, the method of Lagrange multiplier leads to an ill-conditioned matrix which is not positive definite. The aim of this paper is to boost the effectiveness of the conventional reproducing kernel particle method (RKPM) in handling those types of constraints which specify the field variable and its gradient(s) conveniently. Insertion of the gradient term(s), along with generalization of the corrected collocation method, provides a breakthrough remedy in dealing with such controversial constraints. This methodology which is based on these concepts is referred to as gradient RKPM (GRKPM). Since one can easily relate to such types of constraints in the context of beam-columns and plates, some pertinent boundary value problems are analyzed. It is seen that GRKPM, not only enforces constraints and boundary conditions conveniently, but also leads to enhanced accuracy and substantial improvement of the convergence rate.     

Recent advances in the construction of polygonal finite element interpolants

Summary This paper is an overview of recent developments in the construction of finite element interpolants, which areC ⁰-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants—Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.     

Analyzing three-dimensional wave propagation with the hybrid reproducing kernel particle method based on the dimension splitting method

By introducing the dimension splitting method into the reproducing kernel particle method (RKPM), a hybrid reproducing kernel particle method (HRKPM) for solving three-dimensional (3D) wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of two-dimensional (2D) problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the element-free Galerkin method.

     

The Approximation and Computation of a Basis of the Trace Space <Emphasis Type="Italic">H</Emphasis><Superscript>1/2</Superscript>

We present a method for construction of an approximate basis of the trace space H ^1/2 based on a combination of the Steklov spectral method and a finite element approximation. Specifically, we approximate the Steklov eigenfunctions with respect to a particular finite element basis. Then solutions of elliptic boundary value problems with Dirichlet boundary conditions can be efficiently and accurately expanded in the discrete Steklov basis. We provide a reformulation of the discrete Steklov eigenproblem as a generalized eigenproblem that we solve by the implicitly restarted Arnoldi method of ARPACK. We include examples highlighting the computational properties of the proposed method for the solution of elliptic problems on bounded domains using both a conforming bilinear finite element and a non-conforming harmonic finite element. In addition, we document the efficiency of the proposed method by solving a Dirichlet problem for the Laplace equation on a densely perforated domain.     

基于核模糊聚类的动态多子群协作骨干粒子群优化

针对骨干粒子群优化（BBPSO）算法易陷入局部最优、收敛速度低等问题，提出了基于核模糊聚类的动态多子群协作骨干粒子群优化（KFC-MSBPSO）算法。该算法在标准骨干粒子群算法的基础上，首先，采用核模糊聚类方法将主群分割为多个子群，令各个子群协同寻优，提高了算法的搜索效率。然后，引入非线性动态变异因子，根据子群内粒子数以及收敛情况动态调节子群粒子变异概率，通过变异的方式使子群粒子重新回到主群，提高了算法的探索能力；进一步采用主群粒子吸收策略与子群合并策略加强了主群与子群之间、子群与子群之间的信息交流，提高了算法的稳定性。最后，利用子群重建策略，结合主群与子群搜索到的最优解，调节子群重建的间隔代数。通过Sphere等6个标准测试函数进行对比实验，结果表明，KFC-MSBPSO算法和经典BBPSO算法以及反向骨干粒子群优化（OBBPSO）算法等改进算法相比寻优准确率至少提高了约11.1%，在高维解空间内测试结果的最佳均值占到83.33%并且具有更高的收敛速度。这说明KFC-MSBPSO算法具有良好的搜索性能与鲁棒性，可应用于高维复杂函数的优化问题中。     

Explicit 3-D RKPM shape functions in terms of kernel function moments for accelerated computation

The construction of meshless shape functions is more time-consuming than evaluation of FEM shape functions. Therefore, it is of great importance to take measures to speed up the computation of meshless shape functions. 3-D meshless shape functions and their derivatives are, in the context of reproducing kernel particle method (RKPM), expressed explicitly in terms of kernel function moments for the very first time. This avoids solutions of linear algebraic equations and numerical inversions encountered in standard RKPM implementation, thus speeds up computation of meshless shape functions. A numerical test is performed in a hexahedral domain with the mere purpose of comparing the computation time for shape functions construction between the standard RKPM implementation and the enhanced procedure. Then two 3-D elastostatics numerical examples are presented, which demonstrate that the proposed unique treatment of RKPM shape functions is especially effective.     

Looking for the best constant in a Sobolev inequality: a numerical approach

A numerical method for the computation of the best constant in a Sobolev inequality involving the spaces H ²(Ω) and C⁰([`(W)])C^{0}(\overline{\Omega}) is presented. Green’s functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagrangian method, together with finite element approximations, and a Green’s functions based approach. Numerical experiments show the ability of the methods in computing this best constant for various two-dimensional domains, and the remarkable convergence properties of the augmented Lagrangian based iterative method.     

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H ³/h ³) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincaré–Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Optimal maximum norm error estimates for some finite element methods for treating the Dirichlet problem

The goal of this paper is to establish optimalL ^∞ error estimates for a few different finite element type methods for the Dirichlet problem in a bounded domain. The methods are selected so as to avoid the necessity for imposing boundary conditions on the trial functions, usually difficult in practice. Three specific methods are treated. These are the method of interpolated boundary condition and two methods of Nitsche.     

Fully discrete mixed finite element approximations for non-Darcy flows in porous media

A numerical approximation procedure is proposed to solve equations describing non-Darcy flow of a single-phase fluid in a porous medium in two or three spacial dimensions, including the generalized Forchheimer equation. Fully discrete mixed finite element methods are considered and analyzed for the approximation. Existence and uniqueness of the approximation are discussed and optimal order error estimates in L² are derived for the three relevant functions.     

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann–Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n⁺–n–n⁺ diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.     

An immersed interface method for the Vortex-In-Cell algorithm

The paper presents a two-dimensional immersed interface technique for the Vortex-In-Cell (VIC) method for simulation of flows past bodies of complex geometry. The particle–mesh VIC algorithm is augmented by a local particle–particle correction term in a Particle–Particle Particle–Mesh (P³M) context to resolve sub-grid scales incurred by the presence of the immersed interface. The particle–particle correction furthermore allows to disjoin mesh and particle resolution by explicitly resolving sub-grid scales on the particles. This P³M algorithm uses an influence matrix technique to annihilate the anisotropic sub-grid scales and adds an exact particle–particle correction term. Free-space boundary conditions are satisfied through the use of modified Green’s functions in the solution of the Poisson equation for the streamfunction. The concept is extended such as to provide exact velocity predictions on the mesh with free-space boundary conditions.The random walk technique is employed for the diffusion in order to relax the need for a remeshing of the computational elements close to solid boundaries. A novel partial remeshing technique is introduced which only performs remeshing of the vortex elements which are located sufficiently distant from the immersed interfaces, thus maintaining a sufficient spatial representation of the vorticity field.Convergence of the present P³M algorithm is demonstrated for a circular patch of vorticity. The immersed interface technique is applied to the flow past a circular cylinder at a Reynolds number of 3000 and the convergence of the method is demonstrated by a systematic refinement of the spatial parameters. Finally, the flow past a cactus-like geometry is considered to demonstrate the efficient handling of complex bluff body geometries. The simulations offer an insight into physically interesting flow behavior involving a temporarily negative total drag force on the section.     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

The Streamline–Diffusion Method for Conforming and Nonconforming Finite Elements of Lowest Order Applied to Convection–Diffusion Problems

We consider the streamline-diffusion finite element method with finite elements of lowest order for solving convection-diffusion problems. Our investigations cover both conforming and nonconforming finite element approximations on triangular and quadrilateral meshes. Although the considered finite elements are of the same interpolation order their stability and approximation properties are quite different. We give a detailed overview on the stability and the convergence properties in the L ²- and in the streamline–diffusion norm. Numerical experiments show that often the theoretical predictions on the convergence properties are sharp. Received December 7, 1999; revised October 5, 2000