Accuracy tests for various sensitivity analysis methods with respect to shape variables in planar cantilever beams

Sensitivity information is required in the optimal design process. In structural optimization, sensitivity calculation is a bottleneck due to its complexities. Various schemes have been proposed for the calculation. Analytic and finite difference methods are the most popular at the present time; however, they have their advantages and disadvantages. The semi-analytic method has been suggested to overcome these difficulties. In spite of its excellence, the semi-analytic method has been found to possess numerical errors with respect to shape variables. In this research, the errors from each method are evaluated and compared using a shape variable. A planar beam is selected as an example since it has a mathematical solution. An efficient method is suggested for the structural optimization which utilizes the finite element method.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.     

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.     

A study of solution algorithms for shape design sensitivity analysis on a supermini computer with an attached array processor

This paper presents a study and comparison of shape design sensitivity analysis algorithms that are based on the continuum adjoint variable method, the continuum direct differentiation method, and the finite difference method, implemented on a supermini computer with an attached array processor. The basic algorithms and their differences in evaluating shape design sensitivity coefficients are outlined. A solution method for solving a system of equations, using a general sparse storage technique, is used for numerical implementation of shape design sensitivity analysis. It is found that computing shape design sensitivity coefficients using the direct differentiation method is significantly more efficient than using the adjoint variable method or the finite difference method. A detailed performance evaluation of the methods, using an attached array processor, is presented. The performance of the attached array processor, compared to a supermini computer is shown to depend strongly on the type of computations to be carried out. When only parts of a program are running on an attached array processor, the CPU time distribution among the different subroutines of the program can change significantly, compared to using the host processor only.     

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.     

Accuracy test of sensitivity analysis in the semi-analytic method with respect to configuration variables

Configuration optimization is a structural optimization method where the geometrical shape of the structures can be changed during the optimization process. Sensitivity informations are required in the general optimization and quite costly. Especially, they are extemely expensive in the structural optimization where the finite element analysis is utilized. Since the nodal coordinates are regarded as design variables in the configuration optimization, the sensitivities according to the nodal coordinates must be calculated. The characteristics of the configuration optimization is that the transformation matrix in the finite element analysis is a function of design variables. Thus the sensitivity calculation in the configuration optimization is even more complicated. For the efficient sensitivity calculations, various methods have been proposed. They are the analytic method (AM), overall finite difference method (OFD), and semi-analytic method (SM). The semi-analytic method consists of the forward and central difference approximation. This study has been conducted to choose an appropriate method by comparison based on the mathematical and numerical aspects. Some standard structural problems are selected for the evaluations.     

Design sensitivity analysis for transient response of non-viscously damped dynamic systems

A design sensitivity analysis for the transient response of the non-viscously damped dynamic systems is presented. The non-viscously (viscoelastically) damped system is widely used in structural vibration control. The damping forces in the system depend on the past history of motion via convolution integrals. The non-viscos damping is modeled by the generalized Maxwell model. The transient response is calculated with the implicit Newmark time integration scheme. The design sensitivity analysis method of the history dependent system is developed using the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single dof system. The proposed method is also applied to a multi-dof system. The validity and accuracy of the sensitivities from the proposed method are confirmed by finite difference results.     

Second order adjoint sensitivity analysis of multibody systems described by differential–algebraic equations

The optimization strategies employing second order sensitivity information has higher accuracy, but its computation is complex. In this paper, an adjoint variable method applied for the second order design sensitivity analysis of multibody design problems is developed. Based on Lagrange equations of multibody system dynamics, a general objective function, constraint conditions, initial and end conditions, the adjoint variable equations for first order sensitivity analysis and design sensitivity formulations are derived firstly. Then, second order sensitivity analysis formulations, as well as the detailed computation steps, are given based on the previous results. For simplification, the second derivative of the objective function with respect to design variables is translated into an initial value problem of an ordinary differential equation with one variable. Finally, a numerical example of slider–crank mechanism validates the accuracy and efficiency of the method for second order sensitivity analysis.     

Continuum-based sensitivity analysis for coupled atomistic and continuum simulations for 2-D applications using bridging scale decomposition

The concept of linking simulations at multiple length and time scales is found useful for studying local physical phenomena such as crack propagation. Many multi-scale methods, which couple molecular dynamics models with continuum models, have been proposed over the last decade. One of the most advanced methods developed recently is the bridging scale method, in which the total displacement is decomposed into orthogonal coarse and fine scales. This paper presents the continuum-based sensitivity analysis for two-dimensional coupled atomistic and continuum problems using the bridging scale method. A variational formulation for the bridging scale decomposition is developed based on the Hamilton’s principle. The continuum-based variational formulation provides a uniform and generalized system of equations from which the differential equations can be obtained naturally. The sensitivity expressions for both direct differentiation method (DDM) and adjoint variable method (AVM) are derived in a continuum setting. Due to its efficiency for crack problems, the direct differentiation method is chosen to be implemented numerically and applied to two examples, including a crack propagation problem. Both material and sizing design variables are included to reveal the impact of design changes at the macroscopic level to the responses at the atomistic level. Also demonstrated is the feasibility of achieving the variation of the time history kernel computed using numerical procedures. The sensitivity coefficients calculated are shown to be accurate compared with overall finite difference method. The physical implications of the sensitivity results are also discussed, which accurately predict the behavior of the structural responses.     

Shape design sensitivity analysis and optimization of general shape arches

A method is presented for the shape design sensitivity analysis as applied to general arches whose shapes cannot be mapped on one plane. The shape design sensitivity formulation with respect to the perturbation in the direction normal to the middle surface of the shallow arch curve is derived using the material derivative and the adjoint variable method with the system variational equation expressed in a Cartesian coordinate system. A general shape arch is subdivided into segments each of which can be considered as a shallow arch. On each subdivision of the arch, a Cartesian coordinate system is installed and the shape design sensitivity for the shallow arch is applied. For numerical implementation, the finite element method is adopted and each finite element can be considered as such a subdivision. Numerical examples for sensitivity analysis and optimization are presented to illustrate the finite element sensitivity method proposed.     

Design sensitivity analysis and optimization of dynamic response

The paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems. A key feature of the paper is the development of procedures to handle point-wise state variable constraints. Difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints. An alternate treatment of such constraints is proposed, developed and evaluated. In this treatment each point-wise state variable constraint is replaced by several constraints that are imposed at all the local max-points for the original constraint function. The differential equations of motion are formulated in the first-order form so as to handle more general problems. The direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point-wise constraints are presented. With the adjoint variable methods, there are two ways of calculating design sensitivity coefficients. The first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation. Since the adjoint variable methods are better for a large class of problems, an efficient computational algorithm with these methods is presented in detail. Optimum results for several problems are obtained and compared with those available in the literature. The new formulation works extremely well as precise optimum designs are obtained.     

Quasi-wavelet method for time-dependent fractional partial differential equation

In this decade, many new applications in engineering and science are governed by a series of fractional partial differential equations. In this paper, we propose a novel numerical method for a class of time-dependent fractional partial differential equations. The time variable is discretized by using the second order backward differentiation formula scheme, and the quasi-wavelet method is used for spatial discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Numerical examples are obtained to investigate the accuracy and efficiency of the proposed method. The comparisons of the present numerical results with the exact analytical solutions show that the quasi-wavelet method has distinctive local property and can achieve accurate results.     

Computation of the Hessian matrix in aerodynamic inverse design using continuous adjoint formulations

Continuous adjoint formulations for the computation of (first and) second order derivatives of the objective function governing inverse design problems in 2D inviscid flows are presented. These are prerequisites for the use of the very efficient exact Newton method. Four new formulations based on all possible combinations of the direct differentiation method and the continuous adjoint approach to compute the sensitivity derivatives of objective functions, constrained by the flow equations, are presented. They are compared in terms of the expected CPU cost to compute the Hessian of the objective function used in single-objective optimization problems with N degrees of freedom. The less costly among them was selected for further study and tested in inverse design problems solved by means of the Newton method. The selected approach, which will be referred to as the direct-adjoint one, since it performs direct differentiation for the gradient and, then, uses the adjoint approach to compute the Hessian, requires as many as N+2 equivalent flow solutions for each Newton step. The major part of the CPU cost (N equivalent flow solutions) is for the computation of the gradient but, fortunately, this task is directly amenable to parallelization. The method is used to reconstruct ducts or cascade airfoils for a known pressure distribution along their solid boundaries, at inviscid flow conditions. The examined cases aim at demonstrating the accuracy of the proposed method in computing the exact Hessian matrix as well as the efficiency of the exact Newton method as an optimization tool in aerodynamic design.     

Multi-dimensional semi-Lagrangian scheme that guarantees exact conservation

A new numerical method that guarantees exact mass conservation is proposed to solve multi-dimensional hyperbolic equations in semi-Lagrangian form without directional splitting. The method is based on a concept of CIP scheme and keep the many good characteristics of the original CIP scheme. The CIP strategy is applied to the integral form of variable. Although the advection and non-advection terms are separately treated, the mass conservation is kept in a form of spatial profile inside a grid cell. Therefore, it retains various advantages of the semi-Lagrangian schemes with exact conservation that has been beyond the capability of conventional semi-Lagrangian schemes.     

Non-probabilistic reliability-based topology optimization with multidimensional parallelepiped convex model

In this paper, a new non-probabilistic reliability-based topology optimization (NRBTO) method is proposed to account for interval uncertainties considering parametric correlations. Firstly, a reliability index is defined based on a newly developed multidimensional parallelepiped (MP) convex model, and the reliability-based topology optimization problem is formulated to optimize the topology of the structure, to minimize material volume under displacement constraints. Secondly, an efficient decoupling scheme is applied to transform the double-loop NRBTO into a sequential optimization process, using the sequential optimization & reliability assessment (SORA) method associated with the performance measurement approach (PMA). Thirdly, the adjoint variable method is used to obtain the sensitivity information for both uncertain and design variables, and a gradient-based algorithm is employed to solve the optimization problem. Finally, typical numerical examples are used to demonstrate the effectiveness of the proposed topology optimization method.     

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.     

Failure resistant topology optimization of structures using nonlocal elastoplastic-damage model

For energy absorbing structures made up of ductile materials, the plastic strain accumulation often leads to early material damage and failure, which can deteriorate the overall structural performance. The goal of this work is to limit this damage in elastoplastic designs using the density-based topology optimization framework such that the optimized structures can absorb energy in a more controllable manner. To this end, an implicit nonlocal coupled elastoplastic damage model is considered for simulating the material damage and softening behavior. The nonlocal effect from the void elements is removed by introducing a scaling scheme for the nonlocal parameters. Path-dependent sensitivity is derived analytically using an adjoint method whose accuracy is further verified by the central difference method. The effectiveness of the proposed method is demonstrated through several numerical examples. It is shown that the load-carrying capacity, ductility, as well as ultimate plastic work dissipation capacity of the optimized design, can be considerably improved by the proposed method.     

A contact force model considering constant external forces for impact analysis in multibody dynamics

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.