Hierarchical optimization for topology design of multi-material compliant mechanisms

Multi-material topology optimization enables potential design possibilities in the multiphysics and structural designing fields. In this article, a bi-level hierarchical optimization method is introduced to address the multi-material design of compliant mechanisms. The hierarchical optimization develops decomposition approaches allowing the original complex multi-material optimization problem to be reduced to set of low-order single-material optimization sub-problems. The solution of the complex multi-material problem is found as a vector of the single-material sub-problems solutions. All the local sub-problems are solved with the solid isotropic material with penalization method independently, and a stiffness spreading technique is worked out to coordinate components of the global solution of the original problem. Several numerical examples are presented to demonstrate the validity of this method.     

Multi-objective optimization of fuzzy structural systems

It is recognized that there exists a vast amount of fuzzy information in both the objective and constraint functions of the optimum design of structures. Since most practical structural design problems involve several, often conflicting, objectives to be considered, a multi-objective fuzzy programming method is outlined in this work. The fuzzy constraints define a fuzzy feasible domain in the design space and each of the fuzzy objective functions defines the optimum solution by a fuzzy set of points. A method of solving a fuzzy multi-objective structural optimization problem using ordinary single-objective programming techniques is presented. The computational approach is illustrated with two numerical examples.     

Solid-spaced filters: an alternative for narrow-bandpass applications

Solid-spaced filters are composed of one or several thin wafers of excellent optical quality acting as Fabry-Perot spacer layers. We study the different steps of the design and the manufacture of filters following dense-wavelength-division-multiplexing specifications. The design method of such filters requires a tight synergy between numerical simulations and experimental characterizations to correct possible thickness errors. Experimental results of the manufacture and characterization of a three-cavity narrow-bandpass filter and of an interleaver filter are given.     

ON OPTIMAL DESIGN OF MULTI-PURPOSE TIE-BEAMS

The paper presents a method of solving the problem of optimally designing mulli-purpose tie-beams. The problem is that of minimizing the maximum deflection of a simply supported beam under transverse load. The volume of the beam is prescribed and the maximum allowable longitudinal elongation if the beam were to act as a tie is given. The method of solution presented herein makes use of numerical optimization techniques to obtain a design satisfying specified criteria. The solutions thus obtained are compared with the results of others to show the effectiveness and usefulness of the method of solution presented herein.     

Real-time integration and differentiation of analog signals bymeans of digital filtering

A novel filter design method for broadband recursive digital integrators and differentiators is presented. The performance of the digital infinite-impulse response (IIR) filters designed with the method is compared with that of finite-impulse response (FIR) filters and that of classical numerical integration and differentiation. The common conviction that IIR filters with excellent amplitude characteristics always have poor phase behavior is refuted. It is shown that it is possible to design easily realizable IIR integrators and differentiators with an arbitrarily small amplitude and phase error. While there is no FIR alternative for IIR integrators, both FIR and IIR methods give competitive results for differentiators     

Design of a robust thin-film interference filter for erbium-doped fiber amplifier gain equalization

Gain-flattening filters (GFFs) are key wavelength division multiplexing components in fiber-optics telecommunications. Challenging issues in the design of thin-film GFFs were recently the subject of a contest organized at the 2001 Conference on Optical Interference Coatings. The interest and main difficulty of the proposed problem was to minimize the sensitivity of a GFF to simulated fabrication errors. A high-yield solution and its design philosophy are described. The approach used to control the filter robustness is explained and illustrated by numerical results.     

Sensitivity analysis of heat conduction for functionally graded materials

A sensitivity analysis is presented for the steady-state and transient heat conduction of functionally graded materials (FGMs). Based on the finite element method, the sensitivity equations of heat conduction are presented by using the direct method and the adjoint method. In the solution of transient problem, the precise time integration (PTI) is employed. The spatial volume fractions of materials of FGM (size problem) and the shape design parameters are considered. Detailed formulations especial for the FGMs are provided. The numerical examples are presented to demonstrate the precision and applicability of the proposed method.     

挤压成形过程灵敏度分析及模具型腔参数优化设计

以高温下的挤压成形加工过程优化设计问题为背景,针对应变率相关粘性固体本构关系的材料稳态非弹性变形过程提出了形状灵敏度分析方法,并对算法的效率进行了讨论,在此基础上研究了挤压成形模具的形状优化设计问题的数学模型和数值求解算法。给出了挤压成形模具的二维形状优化设计算例,数值结果验证了所给出的灵敏度分析算法和优化设计模型的正确性和可用性。     

The design of a variable-step integrator for the simulation of gas transmission networks

The simulation of a large-scale gas transmission network involves the numerical solution of a large system of initial valued, stiff algebraic/differential equations. Rapid changes in the solution are present due to the disturbances generated by the varying consumer demand and the operation of network controlling devices such as compressors. This paper discusses the design of an efficient variable-step integrator for the solution of the problem. Two sets of strategies are presented for implementing the variable-step integrator; one for the implicit numerical method such as the diagonally implicit Runge–Kutta methods, and the other for the linearly implicit Rosenbrock-type method. The performance of the numerical methods implemented are compared with the British Gas simulation program PAN on a number of large, realistic transmission networks.     

Design and Optimization of airfoils in non-stalling incompressible flow with a prescribed range of the angle of attack

A simple and reliable method of aerodynamic design of airfoils is proposed for a given range of the angle of attack. The method is based on the solution of a new inverse boundary-value problem of aerohydrodynamics. In the problem the initial velocity distributions are given along the contour of the required airfoil as a function of the are co-ordinate for extreme values of the angle of attack from a fixed range. We justify a method of choice of these distributions which guarantees a non-stalling flow around the airfoil. The problem of airfoil optimization is also considered. Results of numerical calculations are presented.     

Geometric approach for designing optical binary amplitude and binary phase-only filters

An analytic solution for real optimal filters is known, and the special case of optimal binary phase-only filters can be solved by a fast binning algorithm but no analytic solution is known. We establish a geometric solution for the design of optimal binary amplitude filters (OBAF's) and optimal binary phase-only filters (OBPOF's) for any object. The optimal filter is found in terms of maximizing the field strength at the origin in the correlation plane. We found that it is possible to construct a unique convex polygon by using an ordered set of phasors from the filter object's Fourier transform. This process leads eventually to an exact solution for the filter-design problem. We show that the maximum distance across the polygon divides the phasors into two groups: For the OBAF, it determines the group that is passed or blocked; for the OBPOF, it determines which group is passed with a zero or a pi phase shift. The shape of the convex polygon gives qualitative information on the criticalness and the tightness needed in the design process. It provides good insight into the binning-process algorithm and permits us to bound the error in the binning process. Design examples through computer simulation and applications in fingerprint identification are presented.     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

An Enhanced Transcribing Scheme for the Numerical Solution of a Class of Optimal Control Problems

An enhanced scheme of transcribing the system dynamics for the numerical solution of optimal control problems is proposed. This new scheme is based on the standard method of direct collocation that converts an optimal control problem into a nonlinear programming problem via simultaneous state and control discretization. When compared with the standard method, the enhanced scheme has the advantage of higher solution accuracy with minimal additional computational effort. It is particularly suited for systems with states that are related to each other in a special form. For such systems, the ensuing nonlinear programming problem has the same number of constraints as those using the standard method. Numerical results on several optimal control problems using the enhanced scheme are presented, together with comparisons with the results obtained from the standard scheme.     

Application of the MFS to inverse obstacle scattering problems

In this paper, the method of fundamental solutions (MFS) is used to detect the shape, size and location of a scatterer embedded in a host acoustic homogeneous medium from scant measurements of the scattered acoustic pressure in the vicinity of the obstacle. A nonlinear constrained minimization regularized MFS technique is proposed for the numerical solution of the inverse problem in question. The stability of the technique is investigated by inverting measurements contaminated by random noise. The results of several numerical experiments are presented.     

Finite element simulations of thin-film composite BAW resonators

A finite element method (FEM) formulation is presented for the numerical solution of the electroelastic equations that govern the linear forced vibrations of piezoelectric media. A harmonic time dependence is assumed. Both of the approaches, that of solving the field problem (harmonic analysis) and that of solving the corresponding eigenvalue problem (modal analysis), are described. A FEM software package has been created from scratch. Important aspects central to the efficient implementation of FEM are explained, such as memory management and solving the generalized piezoelectric eigenvalue problem. Algorithms for reducing the required computer memory through optimization of the matrix profile, as well as Lanczos algorithm for the solution of the eigenvalue problem are linked into the software from external numerical libraries. Our FEM software is applied to detailed numerical modeling of thin-film bulk acoustic wave (BAW) composite resonators. Comparison of results from 2D and full 3D simulations of a resonator are presented. In particular, 3D simulations are used to investigate the effect of the top electrode shape on the resonator electrical response. The validity of the modeling technique is demonstrated by comparing the simulated and measured displacement profiles at several frequencies. The results show that useful information on the performance of the thin-film resonators can be obtained even with relatively coarse meshes and, consequently, moderate computational resources     

Topology optimization in micromechanical resonator design

A?topology optimization problem in micromechanical resonator design is addressed in this paper. The design goal is to control the first several eigen-frequencies of a micromechanical resonator using topology optimization. The design variable is the distribution of mass in a constrained domain which we model via (1)?the Simple Isotropic Material with Penalization Model and (2)?the Peak Function Model. The overall optimization problem is solved using the Method of Moving Asymptotes and a Genetic Algorithm combined with a local gradient method. A?numerical example is presented to highlight the features of the methods in more detail. The advantages and disadvantages of each method are discussed.     

Analysis of multitone holographic interference filters by use of a sparse Hill matrix method

A theory is presented for the application of Hill's matrix method to the calculation of the reflection and transmission spectra of multitone holographic interference filters in which the permittivity is modulated by a sum of repeating functions of arbitrary period. Such filters are important because they may have two or more independent reflection bands. Guidelines are presented for accurately truncating the Hill matrix, and numerical methods are described for finding the exponential coefficient and the coefficients of the Floquet-Bloch waves within the filter. The latter calculation is performed by use of a computational technique known as inverse iteration. The Hill matrix for such problems is sparse, and thus, even though the matrix can be quite large, it may be efficiently stored and processed by a desktop computer. It is shown that the results of using Hill's matrix method are in close agreement with numerical calculations based on thin-film decomposition, a transfer-matrix technique. An important result of this research is the demonstration that Hill's matrix method may, in principle, be used to analyze any multiperiodic problem, so long as the periods are known to finite precision.     

Numerical analysis of 3D magnetostatic fields

Formulations of three-dimensional magnetostatic fields are reviewed for their finite-element analysis. Partial differential equations and boundary conditions are set up for various kinds of potentials. Besides the method using two scalar potentials, several vector potential formulations are also discussed. Galerkin techniques combined with the finite element method are applied for the numerical solution of the boundary value problems. The effect of gauging the vector potential upon the numerical performance is investigated. Solutions by different formulations to a simple test problem and a benchmark problem involving relatively thin saturated iron plates are presented. The latter is compared to measured results.<>     

Design sensitivity analysis and optimization of interface shape for zoned-inhomogeneous thermal conduction problems using boundary integral formulation

A generalized formulation of the shape design sensitivity analysis for two-dimensional steady-state thermal conduction problem as applied to zoned-inhomogeneous solids is presented using the boundary integral and the adjoint variable method. Shape variation of the external and zone-interface boundary is considered. Through an analytical example, it is proved that the derived sensitivity formula coincides with the analytic solution. In numerical implementation, the primal and adjoint problems are solved by the boundary element method. Shape sensitivity is numerically analyzed for a compound cylinder, a thermal diffuser and a cooling fin problem, and its accuracy is compared with that by numerical differentiation. The sensitivity formula derived is incorporated to a nonlinear programming algorithm and optimum shapes are found for the thermal diffuser and the cooling fin problem.