A face based basis function solutions to thermal conduction problems for 3-D irregular shaped bodies using BEM

A shape optimization procedure is developed, using the boundary element method, for two-dimensional anisotropic structures to minimize weight while satisfying certain constraints upon stresses and geometry. A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities of anisotropic materials. The boundary element method is very suitable for shape optimization and in comparison with the finite element method needs much fewer data, related only to the boundary of the structure being considered. Because a directly differentiated form of the boundary integral equation can be used to determine the derivatives of the objective and constraint functions, the accuracy of computation is very high. Because of the non-linear nature of weight and stresses, the numerical optimization method used in the program is the feasible direction approach, together with the golden section method for the one-dimensional search. Three example problems with anisotropic material properties are presented to demonstrate the applications of this general-purpose program.

A previous difference method has a geometrical restriction in analyzing numerically thermal conduction problems about materials with any shape. Therefore it has been impossible to generalize the numerical solution of three-dimensional thermal conduction problems by the previous method. In this paper we propose a dual difference method which is free from the geometrical restriction of the previous method. Thereby the application of difference calculus can be extended to two-dimensional and three-dimentional problems about materials with any shape. The accuracy and the relative error depending on the shape of elements were compared and fount out to be smaller than in the previous method. Then, we made an experiment about the induction heating of a steel ball, and compare the experimental results with calculated results by dual difference method. A close agreement between measured and calculated values was obtained.

We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.

Validity and correctness test verification of the measuring software has been a thorny issue hindering the development of Gear Measuring Instrument (GMI). The main reason is that the software itself is difficult to separate from the rest of the measurement system for independent evaluation. This paper presents a Virtual Gear Measuring Instrument (VGMI) to independently validate the measuring software. The triangular patch model with accurately controlled precision was taken as the virtual workpiece and a universal collision detection model was established. The whole process simulation of workpiece measurement is implemented by VGMI replacing GMI and the measuring software is tested in the proposed virtual environment. Taking involute profile measurement procedure as an example, the validity of the software is evaluated based on the simulation results; meanwhile, experiments using the same measuring software are carried out on the involute master in a GMI. The experiment results indicate a consistency of tooth profile deviation and calibration results, thus verifying the accuracy of gear measuring system which includes the measurement procedures. It is shown that the VGMI presented can be applied in the validation of measuring software, providing a new ideal platform for testing of complex workpiece-measuring software without calibrated artifacts.

Data-driven analysis of parametrized acoustic systems in the frequency domain

A boundary element method is presented for two-dimensional and steady-state analysis of the traveling magnetic field in conductive material. The boundary integral equation is derived from the Green's second identity in which the fundamental solution including an effect due to the motional induction is originated. By use of the constant boundary elements the boundary integral equation is transformed into a system of linear equations. Some numerical results are shown in order to demonstrate the validity of the boundary element method. It is proved that even at large magnetic Reynolds numbers the boundary element solution is stably calculated with the second-order accuracy.

A boundary element formulation, along with detailed solution procedure for determining the acoustic field inside a three-dimensional, rectangular duct is presented in this paper. The results of classical and boundary element solutions are compared for a typical rectangular duct by restricting the input frequency in such a way that only plane wave propagation is possible. The effect of changing the type and number of discrete boundary elements on the computed sound pressure levels inside the duct is also presented.

Elasto-plastic plate bending analysis by a boundary element method with initial plastic moments

On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

This LDRD Senior's Council Project is focused on the development, implementation and evaluation of Reduced Order Models (ROM) for application in the thermal analysis of complex engineering problems. Two basic approaches to developing a ROM for combined thermal conduction and enclosure radiation problems are considered. As a prerequisite to a ROM a fully coupled solution method for conduction/radiation models is required; a parallel implementation is explored for this class of problems. High-fidelity models of large, complex systems are now used routinely to verify design and performance. However, there are applications where the high-fidelity model is too large to be used repetitively in a design mode. One such application is the design of a control system that oversees the functioning of the complex, high-fidelity model. Examples include control systems for manufacturing processes such as brazing and annealing furnaces as well as control systems for the thermal management of optical systems. A reduced order model (ROM) seeks to reduce the number of degrees of freedom needed to represent the overall behavior of the large system without a significant loss in accuracy. The reduction in the number of degrees of freedom of the ROM leads to immediate increases in computational efficiency and allows many design parameters and perturbations to be quickly and effectively evaluated. Reduced order models are routinely used in solid mechanics where techniques such as modal analysis have reached a high state of refinement. Similar techniques have recently been applied in standard thermal conduction problems e.g. though the general use of ROM for heat transfer is not yet widespread. One major difficulty with the development of ROM for general thermal analysis is the need to include the very nonlinear effects of enclosure radiation in many applications. Many ROM methods have considered only linear or mildly nonlinear problems. In the present study a reduced order model is considered for application to the combined problem of thermal conduction and enclosure radiation. The main objective is to develop a procedure that can be implemented in an existing thermal analysis code. The main analysis objective is to allow thermal controller software to be used in the design of a control system for a large optical system that resides with a complex radiation dominated enclosure. In the remainder of this section a brief outline of ROM methods is provided. The following chapter describes the fully coupled conduction/radiation method that is required prior to considering a ROM approach. Considerable effort was expended to implement and test the combined solution method; the ROM project ended shortly after the completion of this milestone and thus the ROM results are incomplete. The report concludes with some observations and recommendations.

The influence of the magnetic configuration on the behavior of magneto-electro-elastic bimorph beams is analyzed by using a boundary element approach. The problem is formulated by using the generalized displacements and generalized tractions. The boundary integral equation formulation is obtained by extending the reciprocity theorem to magneto-electro-elastic problems; it is numerically implemented by using the boundary element method multidomain technique to address problems involving nonhomogeneous configurations. Results under different magnetic configurations are compared highlighting the characteristic features of magnetopiezoelectric behavior particularly focusing on the link between interlaminar stress and magnetic induction.

In chapter 2, we have studied the thermal conduction problem in a plate. This problem can be considered as a simple approach of the more general problem of elasticity. As a matter of fact, in chapter 2, the unknown function was a one component function. Now, in this chapter, we shall consider the elasticity problem where the unknown function is a three component function. There are similarities between problems of thermal conduction and elasticity and in order to intensify these similarities, we shall use the same notations.

Most recently, we have developed a novel multilevel boundary element method (MLBEM) for steady Stokes flows in irregular two-dimensional domains (Grigoriev, M.M., and Dargush, G.F., Comput. Methods. Appl. Mech. Eng., 2005). The multilevel algorithm permitted boundary element solutions with slightly over 16,000 degrees of freedom, for which approximately 40-fold speedups were demonstrated for the fast MLBEM algorithm compared to a conventional Gauss elimination approach. Meanwhile, the sevenfold memory savings were attained for the fast algorithm. This paper extends the MLBEM methodology to dramatically improve the performance of the original multilevel formulation for the steady Stokes flows. For a model problem in an irregular pentagon, we demonstrate that the new MLBEM formulation reduces the CPU times by a factor of nearly 700,000. Meanwhile, the memory requirements are reduced more than 16,000 times. These superior run-time and memory reductions compared to regular boundary element methods are achieved while preserving the accuracy of the boundary element solution.

In [B. Fares et al., J. Comput. Phys., 230 (2011), pp. 5532--5555], a reduced basis method (RBM) for the electric field integral equation (EFIE) using the boundary element method (BEM) is developed, based on a simplified a posteriori error estimator for the greedy-based snapshot selection. In this paper, we extend this work and propose a certified RBM for the EFIE based on a mathematically rigorous a posteriori estimator. A central difficulty of the certified method is that the intrinsic solution space of the EFIE is ${\boldsymbol{H}^{-1/2}_{{\rm div}}(\Gamma)}$, inducing a relatively complicated norm. Since the measured error consists of the difference between the reduced basis solution and the boundary element solution, which is a member of the discrete boundary element space, we clarify that the intrinsic norm can be replaced by an alternative norm and in this work use the $\boldsymbol{H}({\rm div})$-norm, which is computable and demonstrated to not degrade the quality of the error estimator. A successive constraint method (SCM) for complex matrices is discussed in detail, and numerical tests for the SCM and then the certified RBM confirm the analysis.