Design Sensitivity Expressions Research Articles

Reliability-based topology optimization framework of two-dimensional phononic crystal band-gap structures based on interval series expansion and mapping conversion method

• The NRBTO scheme is presented for PnCs where the interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions. • A mapping conversion method is proposed to eliminate the gray-scale elements when relatively coarse grids are used. • The maximum band-gap with a specific frequency region and a filling fraction is obtained while ensuring reliability requirements considering the uncertainty in the optimization parameters. In this study, a non-probabilistic reliability-based topological optimization (NRBTO) method is investigated for the optimized design of the band-gap structure of two-dimensional phononic crystals (PnCs), in which unknown but bounded (UBB) uncertainty issues from the material uncertainty are taken into account. A measurement metric is first defined based on the interval set-theoretical model, which may transform the band-gap setting into a non-probabilistic reliability format. The interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions between frequency measures and design variables. The method of moving asymptote (MMA) is adopted as well to iteratively solve the reliability-oriented band-gap optimization problem for PnCs. To effectively tackle the gray-scale units, a mapping conversion method is involved eventually. The usage and validity of the present methodology are verified through three examples, and numerical results show that considering the uncertainty effect of UBB throughout the topology optimization process has an important impact on the final layout scheme. Additionally, the optimization of the out-of-plane modes of PnCs (both the band-gap upper bound and the filling rate constraint are considered) indicates that the topological configuration may always change accompanied with the filling rate increases.

Isogeometric configuration design optimization of three-dimensional curved beam structures for maximal fundamental frequency

This paper presents a configuration design optimization method for three-dimensional curved beam built-up structures having maximized fundamental eigenfrequency. We develop the method of computation of design velocity field and optimal design of beam structures constrained on a curved surface, where both designs of the embedded beams and the curved surface are simultaneously varied during the optimal design process. A shear-deformable beam model is used in the response analyses of structural vibrations within an isogeometric framework using the NURBS basis functions. An analytical design sensitivity expression of repeated eigenvalues is derived. The developed method is demonstrated through several illustrative examples.

Isogeometric Optimal Design of Compliant Mechanisms Using Finite Deformation Curved Beam Built-Up Structures

Abstract This paper presents a configuration and sizing design optimization method for large deformation planar compliant mechanisms, using a continuum-based adjoint design sensitivity analysis (DSA) approach for built-up structures. Under the total Lagrangian formulation, the Jaumann strain formulation using the discretization of the global displacement field is employed to account for the finite deformation of arbitrarily curved Kirchhoff beams. In multipatch models, a rotational junction continuity condition is imposed using penalty and Lagrange multiplier methods. The developed adjoint DSA method can handle nonconservative loading conditions, which lead to asymmetry of tangent operator. Performance measures are displacements and rotation angles, and neutral axis configuration and cross-sectional thickness are considered as design variables. Also, analytical design sensitivity expressions for the rotation continuity condition are derived. Various compliant mechanisms including path-generators and an angular rotator are synthesized to demonstrate the applicability of the proposed method.

Isogeometric design sensitivity analysis and experimental validation of nanoscale structures considering surface effects

A continuum-based design sensitivity analysis (DSA) method is developed for nanoscale structures with surface effects. To account for the effects of precise geometry in the response and the design sensitivity analyses, we employ an isogeometric approach which uses the same NURBS basis functions as used to describe the geometry of CAD. A direct differentiation method is employed to obtain the analytical design sensitivity using a generalized Young-Laplace equation with high-order surface effects. Effective material properties with the surface effects for silver nanowires are measured from a three-point bending test using atomic force microscopy (AFM). The diameter and the suspended length of silver nanowires are considered as sizing and shape design variables, respectively. The design sensitivity expressions are derived with respect to the design parameters and validated comparing with the experimental results from the AFM scanning, showing an acceptable agreement.

Isogeometric configuration design optimization of heat conduction problems using boundary integral equation

The shape variation of a domain naturally results in both shape and orientation variations, so called configuration variation, when employing a boundary integral equation (BIE) method. A configuration design sensitivity analysis (DSA) method is developed for steady state heat conduction problems using the boundary integral equations in an isogeometric approach, where NURBS basis functions in a CAD system are directly utilized in the response analysis. Thus, we can accomplish a seamless incorporation of exact geometry and the higher continuity into a computational framework. To enhance the accuracy of configuration design sensitivity, the CAD-based higher-order geometric information such as normal and tangent vectors is exactly embedded in the design sensitivity expressions. The necessary velocity field for configuration design obtained from the NURBS is analytically decomposed into shape and orientation velocity fields. It is shown to be essential to consider orientation variations and significant for accurate configuration sensitivity through comparison with finite differencing conventional BIE method. The developed isogeometric configuration DSA method turns out to be accurate compared with the analytic solution and the conventional DSA method. During the optimization, a mesh regularization scheme is employed to avoid excessive mesh distortion, which comes from significant design changes.

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.

Design sensitivity analysis and topology optimization of displacement–loaded non-linear structures

A continuum-based design sensitivity analysis (DSA) method for geometrically nonlinear systems with nonhomogeneous boundary conditions is developed to topologically optimize the displacement–loaded nonlinear structures. In the adjoint variable method, the solution space requires just homogeneous boundary conditions even if the original system has nonhomogeneous ones. A design sensitivity expression for the instantaneous rigidity functional is derived for the displacement–loaded nonlinear topology optimization. The tangent stiffness is obtained at the end of the equilibrium iterations in the nonlinear analysis of the original system; this stiffness is used in the DSA so that no iteration would be necessary to evaluate the design sensitivity expressions. In force–loaded systems, the solution dose not converge easily because the material distributes sparsely sometimes during optimization. However, when the displacement–loaded system is used, there is no convergence difficulty.

DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION OF AN ENGINE MOUNT SYSTEM USING AN FRF-BASED SUBSTRUCTURING METHOD

The FRF-based substructuring method is one of the most powerful methods in analyzing the responses of complex built-up structures with high modal density. In this paper, a general procedure for the design sensitivity analysis of a vibro-acoustic system has been presented using the FRF-based substructuring formulation. For an acoustic response function, the proposed method gives a parametric design sensitivity expression in terms of the partial derivatives of the connection element properties and the transfer functions of the substructures. The derived noise sensitivity formula is combined with a non-linear programming module to obtain the optimal design for the engine mount system of a passenger car. The objective function is defined as the area of the interior noise graph integrated over a concerned r.p.m. range. The interior noise variations with respect to the dynamic characteristics of the engine mounts and bushings have been calculated using the proposed sensitivity formulation and transferred to a non-linear optimization software. To obtain the FRFs, a finite element analysis was used for the engine mount structures and experimental techniques were used for the trimmed body including the cabin cavity. The optimization based on the sensitivity analysis gives the ideal stiffness of the engine mount and bushings. The resultant interior noise in the passenger car shows that the proposed method is efficient and accurate.

SENSITIVITY ANALYSIS IN THERMO-ELASTIC-PLASTIC PROBLEMS

The design sensitivity of a thermo-elastic-plastic system is considered. The continuum approach and resulting finite element formulations are described. Sensitivity is obtained by an incremental load approach for nonlinear response analysis. Kinematic and isotropic hardening in thermo-elasto-plasticity are discussed. The control volume approach is to analyze shape and nonshape design. The direct differentiation method is adopted to obtain the design sensitivity expression for the response variables. Analytical examples demonstrate the developed sensitivity procedure. The design sensitivity formulas are discretized to implement it as a computer program with isoparametric finite elements. Numerical examples are presented to calculate the design sensitivity.

Derivation of a general sensitivity formula for shape optimization of 2-D magnetostatic systems by continuum approach

This paper presents a generalized sensitivity formula that is applicable to any objective function of the optimization problems in two-dimensional (2-D) linear magnetostatic systems. For the objective function defined on an interface boundary as well as in a region, the shape design sensitivity expression is derived in a closed form from the augmented Lagrangian method, the material derivative concept and the adjoint variable method. The resultant sensitivity formula is expressed as a line integral along the interface boundary undergoing shape modification. Three shape design problems, whose objective functions are defined on an interface boundary or in a region, are tested to validate the derived sensitivity formula.

Shape design optimization of joining mechanism using doubly curved shell

Methods for shape and topology design sensitivity analysis (DSA) and optimization of joining mechanisms, such as spot welding and adhesive bonding, using a doubly curved (Hughes–Liu) shell are presented in this article. The material derivative concept of continuum mechanics and the adjoint variable method are used to derive the shape and topology design sensitivity expressions in the domain integral form. The shape optimization is performed to find the optimum locations of spot welds to maximize the stiffness of the structure. This optimum design is used as the initial design to perform topology optimization based on a density approach to study the effects of material distribution on structural performances. A numerical example, by postprocessing NIKE3D finite element analysis (FEA) code, of shape and topology optimization is given to demonstrate the validity of the methodologies presented in this study.

Efficient redesign of damped large structural systems via domain decomposition with exact dynamic condensation

A new redesign technique via domain decomposition with exact dynamic condensation is proposed to enhance computational efficiency of redesign of damped large structural systems. A structural system is regarded as an assemblage of substructures. Nodes in each substructure are classified into boundary nodes and interior nodes. The design variables are determined so that the undamped fundamental natural frequency of the total system and deformation component ratios in the resonant damped steady-state vibration would attain the target values. It is shown that new design sensitivity expressions of the lowest eigenvalue (square of the fundamental circular frequency) can be derived in terms of a condensed coordinate system including boundary nodes only. Incremental inverse problem formulation is then applied to the model in terms of the condensed coordinate system. Different from the conventional static domain decomposition method within the context of redesign, difficulties resulting from inertial force terms in eigenvalue problems and forced steady-state vibration problems are resolved. The validity and order of approximation of the proposed method are demonstrated through an example model.

Continuum shape design sensitivity analysis of magnetostatic field using finite element method

A shape design sensitivity analysis for the magnetostatic field is developed using the finite element method. The design sensitivity expressions are analytically derived using the continuum approach, the domain integration method, and the adjoint variable method. A computer program, MAGSEN is developed using object orient programming and validated by applying it to the shape optimization of 8-pole 12-slot BLDC spindle motor.

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.

Toward a Practical Design Optimization Tool

Abstract: Design optimization methodologies for sizing and shape optimal design have undergone an evolutionary process over the last two decades. The motivation behind the changes is to develop a methodology with no practical limitations on the nature and size of the problem that can be solved. In this research, a personal computer‐based shape optimal design system is developed with particular attention to robustness, generality, efficiency, and pre‐ and postprocessing. To achieve these objectives, the hybrid natural shape optimization approach is developed and used with adequate control over the velocity field matrix generation. User control over model creation and mesh generation makes it possible to minimize mesh distortion and the need for remeshing. Design sensitivity expressions using the implicit differentiation method are derived. It is shown that the analytical derivatives can be computed efficiently for a variety of functions. Numerical examples are solved to illustrate the developed methodology using a software system developed under the MS‐DOS operating system.

Material derivative and control volume approaches to shape sensitivity analysis of nonlinear transient thermal problems

The problem of the shape design sensitivity of nonlinear transient thermal systems is considered. The nonlinear shape design sensitivity is formulated by using a weighted residual method employing the continuum description. Both material derivative and control volume approaches for structural shape design sensitivity analysis are presented, analysed and compared. It is shown that the final design sensitivity expressions for the two approaches are theoretically equivalent.

Configuration Design Sensitivity Analysis of Nonlinear Structural Systems with Elastic Material*

ABSTRACT A continuum-based design sensitivity analysis (DSA) method is presented for configuration design of nonlinear structural systems using Mindlin plate and Tim-oshenko beam theories. Both displacement and critical load performance measures are considered. Configuration design variables are characterized by shape and orientation changes of structural components. The material derivative that is used to develop the continuum-based shape DSA method is extended to account for effects of configuration design variation. The piecewise linear design velocity field, i.e., C0-regular, is used to support configuration design changes for a broad class of built-up structures with beams and plates. To allow use of the C0-design velocity field, mathematical models of beam and plate bending must be second-order partial differential equations, so that only first-order derivatives appear in the integrand of the energy equation and, thus, in the integrand of the configuration design sensitivity expression. Since the Mindlin plate and Timoshenko beam theories use displacement and rotation to describe structural response, mathematical models of beam and plate bending are reduced to second-order partial differential equations. The isoparametric finite element formulations are used for numerical evaluation of continuum design sensitivity expressions.

A computational method for design sensitivty analysis of elastoplastic structures

Design sensitivity analysis of structural systems having elastoplastic material behavior is developed using the continuum formulation and its implementation into a finite element program is described. The main idea is to develop a computational procedure for design sensitivity analysis based on the response obtained by an incremental load approach for non-linear response analysis, where the elastoplatic (time-independent) constitutive model is employed to account for the plastic material behavior; i.e. kinematic hardening and isotropic hardening. Discontinuities in the design sensitivity coefficients are investigated and a procedure is proposed to alleviate them. The control/reference domain approach is used to treat shape and non-shape design variables simultaneously. The direct variation method is adopted for this path-dependent problem to obtain the design sensitivity expression for the response variables. Analytical examples are used to demonstrate the developed sensitivity procedure and gain insights for numerical implementation. The design sensitivity expression is discretized to implement it into a computer program using the isoparametric formulation of finite element analysis. Numerical examples are presented to calculate the design sensitivity coefficients and compare them with those using the central finite difference method.

Optimal design of nonlinear parabolic systems. Part II: Variable spatial domain with applications to casting optimization

In Part I, design sensitivity expressions for nonlinear parabolic systems were derived and then specialized for the analysis for thermal systems which govern many processing operations. However, that presentation did not account for shape variations. Here we present the shape sensitivity analysis for the thermal system. The sensitivities are derived via the domain parameterization method and computed with the finite element method. Various means for parameterizing the shape are also discussed. Finally, the sensitivity analysis is coupled with nonlinear programming to optimize the design of a sand casting.

Tangent operators, sensitivity expressions, and optimal design of non-linear elastica in contact with applications to beams

The tangent operator and design sensitivity expressions for non-linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method. The sensitivity computations are then combined with a numerical programming package to create an optimal design environment. To exemplify the optimal design environment, a beam's contact surface is contoured to minimize bending stress. This work combines the research on the sensitivity analysis of non-linear elastic bodies subject to constraints, finite strain non-linear elastic beam analysis and contact analyses. The analysis is valid for any smooth contact surface; and specialized for the case in which the surface is represented by a cubic spline. The direct differentiation method is utilized to perform the sensitivity analysis. In an example problem, the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve.