Hamilton-Jacobi equations are frequently encountered in applications,e.g.,in control theory and differential games.Hamilton-Jacobi equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter.Similarity also exists for the multidimensional cases,and this is helpful in designing difference approximations.In this paper central weighted essentially non-oscillatory (CWENO) schemes for Hamilton-Jacobi equations are investigated,which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives.The schemes are numerically tested on a variety of one-dimensional problems,including a problem related to control optimization.High-order accuracy in smooth regions,high resolution of discontinuities in the derivatives,and convergence to viscosity solutions are observed.

High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations

This page shows some examples of multimedia files. It is also available at: http://www.ieee-uffc.org/tr/mexample.pdf. For submission of multimedia manuscripts to TUFFC, please follow “Information for Contributors” at: http://www.ieeeuffc.org/tr/contrib.pdf. Multimedia Example Created by Jian-yu Lu, Editor-in-Chief, 07/07/03 IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society 1. Color Figure: The IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control (TUFFC) now accepts color figures online with corresponding grayscale figures in print without additional charges if authors follow the multimedia manuscript submission procedure in the “Information for Contributors”. The “color figures” icon below indicates in the print that a color version of the figure is available online. It also links to the color figure submitted originally by the authors for viewing details. Because of the sizes of color figures and the additional editorial work such as making two sets of PostScript files in which they remain identical after replacing grayscale with color, authors should request color figures online only when it is necessary. The color figure must be in JPEG (Joint Photographic Expert Group) or GIF (Graphics Interchange Format) format to reduce file size and to decrease the bandwidth usage on the TUFFC server. Fig. 1. An eight-layer printed circuit board (PCB) used primarily for multi-channel ultrasound signal reception and storage. The board was designed in the Ultrasound Lab at The University of Toledo, led by Dr. Jian-yu Lu. It is one of the many PCBs designed for a high-frame rate medical ultrasound imaging system [1], which consists of 128 linear, ±144V peak, 0.05-10MHz, and 12-bit arbitrary waveform generators for the production of ultrasound signals or for other research purposes. 2. Movies: Please click on the movie icon to see a movie. The two movies below give you an idea on the file size versus movie quality. “VCD quality” here means 1150kbits/s for video and 224kbits/s for 16-bit MP2 stereo audio at 44.1KHz sampling rate. Fig. 2. An introduction to the Ultrasound Lab at The University of Toledo. 3. Movies and Animations: Please click on the movie icons to see movies or animations. Fig. 3. Circuit design and construction of the imaging system. 4. Movies and Animation: Please click on the movie icons to see movies or animation. Movie [File size: 785KB; Format: MPEG1; Resolution: 320X240; Duration: 9 seconds]

IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control

Probabilistic interval analysis for structures with uncertainty

A multi-resolution approach for multi-material topology optimization based on isogeometric analysis

Hybrid uncertain static analysis with random and interval fields

A level-set based multi-material topology optimization method using a reaction diffusion equation is proposed in this paper. Each phase is represented by a combined formulation of different level set functions. This description model is modified from Multi-Material Level Set (MM-LS) topology description model. With a total number of M level set functions, this approach provides a representation of M materials and one void phase (totally M + 1 phases). By this approach, the mathematic model of the multi-material topology optimization problem using a reaction diffusion equation is established. With this model, the geometrical complexity of optimal solutions can be easily controlled by appropriately setting a regularization parameter. Some implementation details for solving this model are also presented in this paper. Finally, several typical numerical examples are shown to confirm the effectiveness of the proposed method. A multi-material topology optimization method using a reaction diffusion equation.Modified Multi-material Level Set (MM-LS) topology description model.Controllable geometrical complexity of optimal solutions.

A level-set based multi-material topology optimization method using a reaction diffusion equation

The problem of dynamic analysis of truss structures based on probability is studied in this paper. Considering the randomness of both physical parameters (elastic module and mass density) of structural materials and geometric dimension of bars respectively or simultaneously, the stiffness and mass matrixes of the elements and structure have been built. The structure dynamic characteristic based on probability is analyzed, and the expressions of numeral characteristics of inherence frequency random variable are derived from the Rayleigh`s quotient. The method of structural dynamic analysis based on probability is developed. Finally, two examples are given.

Probabilistic dynamic analysis of truss structures

Dynamic response analysis of linear stochastic truss structures under stationary random excitation

In this work, a meshless method based on the alternating active-phase algorithm is proposed for the multi-material topology optimization problems. Mathematic model of the proposed method is built by the solid isotropic microstructure with penalization (SIMP) theory and solved by the optimality criteria. During the optimization process, the nodal relative density is chosen as the design variable and Shepard interpolation combined with the moving least squares (MLS) shape function is utilized to obtain the nodal relative density. Nodal integration method is then adopted to obtain the structural stiffness matrix, with the purpose of promoting the computational efficiency. Since the element-free Galerkin (EFG) method is applied to analyze the structure, sensitivity filtering is avoided and mesh-dependence phenomena are alleviated. Several numerical examples are provided to illustrate the validity and feasibility of the proposed method.

A meshless method for multi-material topology optimization based on the alternating active-phase algorithm

A multi-material proportional topology optimization (PTO) method based on the modified material interpolation scheme is proposed in this work. PTO method is a highly heuristic algorithm by which satisfactory results are obtained. When the proposed method is used to solve the minimum compliance problem, the design variables are assigned to elements proportionally by the value of compliance during the optimization process. It is worth mentioning that PTO algorithm does not incorporate sensitivities. Accordingly, there is nothing about sensitivity calculation but just a weighted density used as filtering in the proposed method. Hence, non-sensitivity is also one of the salient features of this method. According to the characteristics, a density interpolation approach based on the logistic function is introduced in the present study. This approach cannot only establish the relationship between the material densities and Young’s modulus more reasonably, but also effectively realize the polarization of the intermediate-density elements. The complication associated with sensitivities can be avoided by the complicated interpolation scheme in conjunction with PTO algorithm. The multi-material interpolation scheme is modified from the extended SIMP interpolation approach in three-phase topology optimization. A density-filter-based Heaviside threshold function combined with the modified interpolation is introduced in this work to obtain clear 0/1 optimal topology design. The effectiveness and feasibility of the proposed method are demonstrated by several typical numerical examples of multi-material topology optimization, in which the optimal design with distinct boundaries can be obtained.

Mingtao Cui

Papers

A level-set based multi-material topology optimization method using a reaction diffusion equation

Probabilistic dynamic analysis of truss structures

Dynamic response analysis of linear stochastic truss structures under stationary random excitation

A meshless method for multi-material topology optimization based on the alternating active-phase algorithm

Multi-material proportional topology optimization based on the modified interpolation scheme