Analytical Design Sensitivity Analysis Research Articles

Shape optimization with immersed interface finite element method

AbstractShape Optimization using conventional interface‐fitted finite element method requires either mesh morphing or remeshing, which is computationally expensive and prone to errors due to suboptimal mesh or solution interpolation. In this article, we present a shape optimization scheme which uses a four‐noded rectangular immersed‐interface element circumventing the need for interface‐fitted mesh for finite element analysis, and mesh morphing or re‐meshing for shape optimization. The analysis problem is solved using a non‐conformal, Petrov–Galerkin (ncPG) formulation. We use non‐conformal trial functions with conformal test functions. A fixed structured grid with a linear approximation for the interface within each element is used. We perform a bi‐material patch test to confirm the consistency and convergence of the immersed‐interface finite element method (IIFEM). The convergence of the displacement and stress error norms are of the same or slightly better order compared to the interface‐fitted FEM. Thus, IIFEM reduces the cost of meshing without compromising the accuracy of the solutions obtained. We then perform analytical design sensitivity analysis to obtain the gradient of the global stiffness matrix with respect to shape design variables. The sensitivities are used in the gradient‐based optimization formulation of the shape design problem. We present several ways to parametrize the design space. Finally, verification and case studies are presented to demonstrate the accuracy and potential of the approach.

A modified extended finite element method approach for design sensitivity analysis

SummaryThis paper describes a modified extended finite element method (XFEM) approach, which is designed to ease the challenge of an analytical design sensitivity analysis in the framework of structural optimisation. This novel formulation, furthermore labelled YFEM, combines the well‐known XFEM enhancement functions with a local sub‐meshing strategy using standard finite elements. It deviates slightly from the XFEM path only at one significant point but thus allows to use already derived residual vectors as well as stiffness and pseudo load matrices to assemble the desired information on cut elements without tedious and error‐prone re‐work of already performed derivations and implementations. The strategy is applied to sensitivity analysis of interface problems combining areas with different linear elastic material properties. Copyright © 2015 John Wiley & Sons, Ltd.

Density-based Topology Design Optimization ofPiezoelectric Crystal Resonators

Abstract Design sensitivity analysis and topology design optimization for a piezoelectric crystal resonator are developed. The piezoelectric crystal resonator is deformed mechanically when subjected to electric charge on the electrodes, or vice versa. The Mindlin plate theory with higher-order interpolations along thickness direction is employed for analyzing the thickness-shear vibrations of the crystal resonator. Thin electrode plates are masked on the top and bottom layers of the crystal plate in order to enforce to vibrate it or detect electric signals. Although the electrode is very thin, its weight and shape could change the performance of the resonators. Thus, the design variables are the bulk material densities corresponding to the mass of masking electrode plates. An optimization problem is formulated to find the optimal topology of electrodes, maximizing the thickness-shear contribution of strain energy at the desired motion and restricting the allowable volume and area of masking plates. The necessary design gradients for the thickness- shear frequency(eigenvalue) and the corresponding mode shape(eigenvector) are computed very efficiently and accurately using the analytical design sensitivity analysis method using the eigenvector expansion concept. Through some demonstrative numerical examples, the design sensitivity analysis method is verified to be very efficient and accurate by comparing with the finite difference method. It is also observed that the optimal electrode design yields an improved mode shape and thickness-shear energy.

Integrated topology optimization with embedded movable holes based on combined description by material density and level sets

This paper presents a novel topology description model for topology optimization problems with embedded movable holes, combining the ability of the level set model for accurate geometrical description of the prescribed hole shapes, and the high efficiency of the material density-based method. By this means, any arbitrary hole shapes can be represented accurately and smoothly, while the topological changes can be easily handled by the material distribution model. An explicit mathematical expression is defined to obtain the actual structural layout by combing both models. Moreover, an effective model for the non-overlap constraint is proposed in a unified and systematic manner to avoid the overlaps between the holes and between each hole and the design domain boundary. Therein, the non-overlap constraint for all the embedded holes is imposed as a single explicit integral constraint over the design domain, thus avoiding the difficulties in the overlap detection of multiple arbitrary-shaped geometries. Such a non-overlap constraint is accurate and differentiable, facilitating an analytical design sensitivity analysis. Numerical examples are given to demonstrate the effectiveness and efficiency of the present method.

Analytical DSA for Explicit Dynamics of Elastic-plastic Shells

The paper presents an analytical constitutive design sensitivity analysis (DSA) algorithm for explicit dynamics of elastic-plastic finite rotation shells. Two explicit dynamical algorithms for finite rotation shells are presented, and the DSA is developed for the one formulated in terms of the rotation vector and its time derivatives, \(\{ \mathbf{\psi}, \mathbf{\dot{\psi}}, \mathbf{\ddot{\psi}}\}\). The hypo-elastic constitutive model based on the Green–McInnis–Naghdi stress rate is used to derive an incremental algorithm in terms of ‘back-rotated’ objects. The associative deviatoric Huber–Mises plasticity modified by plane stress conditions is implemented in the form suitable for finite rotation/small elastic strain increments. The analytical DSA is developed for the above-specified problem, with the design derivatives calculated w.r.t. material parameters. Design- differentiation of the dynamic algorithm and the scheme of handling the history data and the predicted values in differentiation, which is crucial in computing correct derivatives, are described. Besides, we show how to avoid Newton loops in the DSA algorithm, when such a loop is present in the constitutive algorithm. Numerical examples show that, despite a great complexity of the solution algorithm for the finite-rotation elastic-plastic shells, it is feasible to compute analytical design derivatives of very good accuracy.

준해석적 비선형 설계민감도를 위한 보정변위하중법

Three methods for design sensitivity such as numerical differentiation, analytical method and semianalytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semianalytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

Topology Designs of Slider Air Bearings

A new approach for topology designs of slider air bearings in magnetic recording disk drives is suggested by using large-scale discrete variable optimization techniques. Conventional optimization techniques are restricted to the original topology of the slider by modifying the initial designs. To overcome the restriction, a new topology design approach is presented with enhanced mathematical techniques. Topology optimization of slider air bearings typically has a large number of design variables because the finite mesh must be fine enough to represent the shape of the air bearing surface (ABS). To handle a large number of design variables, an efficient strategy for the optimization including the sensitivity analysis must be established. As a gradient-based local optimization algorithm, the sequential unconstrained minimization technique (SUMT) using an exterior penalty function is used, which requires little computational effort and computer memory. For the gradient calculation, the analytical design sensitivity analysis method introducing an adjoint variable is employed. A topology design problem is formulated as a function of the residuals which is calculated by solving the generalized Reynolds equation. A very large number of discrete design variables (=9409) are dealt with, which denote the rail heights at grid cells. To validate the suggested design methodology, a developed program is applied to two slider models with one and three trailing rails. The simulation results demonstrated the effectiveness of the proposed design methodology by showing that the optimized topologies have reasonable shapes without any initial designs.

Adjoint Design Sensitivity Analysis of Molecular Gas Film Lubrication Sliders

This paper proposes an analytical design sensitivity analysis (DSA) to topological parameters of MGL (molecular gas film lubrication) sliders by introducing an adjoint variable method. For the analysis of slider air bearings, we used the spatial discretization of the generalized lubrication equation based on a control volume formulation. The residual functions for inverse analysis of the slider are considered as the equality constraint functions. The slider rail heights of all grid cells are chosen as design variables since they are the topological parameters determining air bearing surface (ABS). Then, a complicated adjoint variable equation is formulated to directly handle the highly nonlinear asymmetric coefficient matrix and vector in the discrete system equations of slider air bearings. An alternating direction implicit (ADI) scheme is utilized to efficiently solve large-scale problem in special band storage. The simulation results of DSA are directly compared with those of finite-difference approximation (FDA) to show the effectiveness and accuracy of the proposed approach. The overall sensitivity distribution over the ABS is reported, and clearly shows to which section of the ABS the special attention should be given during the manufacturing process. It is demonstrated that the proposed method can reduce more than 99 percent of the CPU time in comparison with FDA, even though both methods give the same results.

Local analytical design sensitivity analysis of the forging problem using FEM

The present paper concerns the development of a new Local Analytical Sensitivity Analysis (LASA) for calculating the design sensitivities of the forging problem response. The proposed approach is based on the differentiation with respect to the design parameters of each local equation of the highly nonlinear mechanical problem. Thus, the desired design sensitivities are solutions to the finite element resolution of the sensitivity problem corresponding to this differentiation. The justification of the mathematical formalism of the proposed local analytical sensitivity analysis (LASA) has been developed by considering an open flat die forging of an incompressible cylinder. The numerical implementation has been achieved with finite element Local Sensitivity Analysis Software (LSAS). The first part concerns only the first-step forging operation. For this case, the sensitivity results obtained with LSAS are very satisfactory compared to the analytical solutions or/and to the centered finite differences (CFD) computations. The second part consists of associating the incremental character of the forging problem resolution with the proposed LASA approach. The gradient computation of the dissipated strain energy with respect to the design parameter is presented by considering the previous problem with frictionless sliding contact. The numerical gradients obtained by using the LASA approach or/and the CFD computations are compared to the ones calculated from the analytical sensitivities. Comparing to the CFD method calculation ( three highly nonlinear computations), the LASA associated with the incremental character of the forging problem results ( one linear computation) achieves to best accuracy along with significant improvements in the required sensitivity computing time.

A general optimization strategy for sound power minimization

A general approach for minimizing radiated acoustic power of a baffled plate excited by broad band harmonic excitation is given. The steps involve a finite element discretization for expressing acoustic power and vibration analysis, analytical design sensitivity analysis, and the use of gradient-based optimization algorithms. Acoustic power expressions are derived from the Rayleigh integral for plates. A general methodology is developed for computing design sensitivities using analytical expressions. Results show that analytical sensitivity analysis is important from both computational time and accuracy considerations. Applications of the optimization strategy to rectangular plates and an engine cover plate are presented. Thicknesses are chosen as design variables.