The ICM method with objective function transformed by variable discrete condition for continuum structure

ICM (Independent Continuous Mapping) method can solve topological optimization problems with the minimized weight as the objective and subjected to displacement constraints. To get a clearer topological configuration, by introducing the discrete condition of topological variables and integrating with the original objective, an optimal model with multi-objectives is formulated to make the topological variables approach 0 or 1 as near as possible, and the model reduces the effect of deleting rate on the result. The image-filtering method is employed to eliminate the checkerboard patterns and mesh dependence that occurred in the topology optimization of a continuum structure. The computational efficiency is enhanced through selecting quasi-active displacement constraints and a design region. Numerical examples indicate that this algorithm is robust and practicable, though the number of iterations is slightly increased with respect to the original algorithm. The project supported by the National Natural Science Foundation of China (10472003), Beijing Natural Science (3002002) and Beijing Educational Committee Foundations (KM200410005019) and Suspensoried by American MSC Company. The English text was polished by Keren Wang.     

球形阵列自适应网格分层定位算法

刚性球形传声器阵列可以在无空间模糊的条件下进行球谐域数据处理,基于球形阵列的可控波束形成器导向响应功率(SRP)算法定位精度高,但是计算量大,计算效率低。通过将球面致密的全局网格搜索替换为分层搜索策略可以有效减小SRP算法的计算量。提出MRE-SRP算法保持球形阵列SRP定位精度的同时降低计算量,首先通过球谐域MUSIC(SH-MUSIC)算法判断入射声源的数量减小搜索区域;其次将相对熵模型引入球谐域SRP(SH-SRP)定位算法中,提取网格分层前后的信息增益,设计自适应网格选择判据,实现分层多分辨率网格的精准再细分,从而降低计算量。实验验证了所提出算法的性能,结果显示在单双声源定位中,该算法可以实现较高的定位精度,精准选择分层网格,计算量减少75%以上。     

Pointwise convergence of bounded cascade sequences

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function , a cascade sequence is constructed by the iteration where is defined by In this paper, under a condition that the sequence is bounded in , we prove that the following three statements are equivalent: (i) converges . (ii) For , there exist a positive constant and a constant such that (iii) For some converges in . An example is presented to illustrate our result.

     

A fast and practical method to pack spheres for mesh generation

Sphere packing is an attractive way to generate high quality mesh. Several algorithms have been proposed in this topic, however these algorithms are not sufficiently fast for large scale problems. The paper presents an efficient sphere packing algorithm which is much faster and appears to be the most practical among all sphere packing methods presented so far for mesh generation. The algorithm packs spheres inside a domain using advancing front method. High efficiency has resulted from a concept of 4R measure, which localizes all the computations involved in the whole sphere packing process.     

A construction of interpolating wavelets on invariant sets

We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.

     

Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions

The numerical solution of linear elliptic partial differential equations often involves finite element discretization, where the discretized system is usually solved by some conjugate gradient method. The crucial point in the solution of the obtained discretized system is a reliable preconditioning, that is to keep the condition number of the systems under control, no matter how the mesh parameter is chosen. The PCG method is applied to solving convection-diffusion equations with nonhomogeneous mixed boundary conditions. Using the approach of equivalent and compact-equivalent operators in Hilbert space, it is shown that for a wide class of elliptic problems the superlinear convergence of the obtained preconditioned CGM is mesh independent under FEM discretization.     

A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose–Einstein condensates

Numerical computations of stationary states of fast-rotating Bose–Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method with mesh adaptivity by metric control, as an alternative approach to the commonly used high-order (finite difference or spectral) approximation methods. The mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states: an imaginary time propagation method and a Sobolev gradient descent method. We first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful. Mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases. Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state. Considerable computational time saving is obtained compared to uniform mesh computations. The new method is applied to compute difficult cases relevant for physical experiments (large nonlinear interaction constant and high rotation rates).     

Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers

A parallel approach to solve three-dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a Lagrange multiplier technique is presented. The strategy is based on non-overlapping domain decomposition methods, and Lagrange multipliers are used to enforce continuity at the boundaries between subdomains. The novelty of the work is the coupled approach for solving the velocity–pressure-Lagrange multiplier algebraic system of the discrete Navier–Stokes equations by a distributed memory parallel ILU (0) preconditioned Krylov method. A penalty function on the interface constraints equations is introduced to avoid the failure of the ILU factorization algorithm. To ensure portability of the code, a message based memory distributed model with MPI is employed. The method has been tested over different benchmark cases such as the lid-driven cavity and pipe flow with unstructured tetrahedral grids. It is found that the partition algorithm and the order of the physical variables are central to parallelization performance. A speed-up in the range of 5–13 is obtained with 16 processors. Finally, the algorithm is tested over an industrial case using up to 128 processors. In considering the literature, the obtained speed-ups on distributed and shared memory computers are found very competitive.     

Regularity of refinable functions with exponentially decaying masks

The regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus. The main results of this paper are really extensions of some results in Cohen et al. (1999) [5], Jia (1999) [17] and Lorentz and Oswald (2000) [28].     

Convergence of cascade algorithms by frequency approach

Starting with an initial function ?₀, the cascade algorithm generates a sequence by cascade operator Qa defined by