Design Sensitivity Analysis Method Research Articles

Accurate and efficient boundary method for isogeometric shape design sensitivity analysis considering tangential divergence of non-smooth boundary

Accurate and efficient boundary method for isogeometric shape design sensitivity analysis considering tangential divergence of non-smooth boundary

Simultaneous analysis and design optimization for seismic retrofitting of hysteretic structures with fluid viscous dampers

Fluid viscous dampers, if properly sized and placed, can significantly improve the performance and safety of existing structures during an earthquake. This has motivated the development of several approaches to optimize the design of fluid viscous dampers. The optimization approaches available typically have an iterative nature, and in every iteration, expensive time-history analyses are performed. This paper presents a novel optimization-based approach for retrofitting hysteretic structures with fluid viscous dampers that does not require the computation of time-history analyses. The sizing and placement of the dampers are defined using a simultaneous analysis and design optimization approach based on nonlinear programming. In this approach, the equations of motion are treated as equality constraints of the optimization problem. As a consequence, design sensitivity analysis methods are not needed to calculate the gradients of the objective and constraint functions. Two design scenarios are considered: structures with hysteretic behavior retrofitted with linear or nonlinear fluid viscous dampers. The optimization variables include the damping coefficients of the dampers and the structural response over time. The manufacturing cost of the dampers is minimized, with constraints on the inter-story drifts. A continuation scheme on the design-driving inter-story drift constraints is proposed to effectively solve the class of problems considered. The numerical results demonstrate that the proposed approach can solve complex design optimization problems using reasonable computational resources and time. It is also shown that, thanks to the proposed approach, additional constraints can be incorporated in the problem formulation with minimal modeling effort, and without the need to develop additional sensitivity analyses for computing the gradients of the added constraints. The Julia codes used in this study are freely available on GitHub: https://github.com/pollinico/SAND_opt_nonlinFVDs.

An integrated design method for used product remanufacturing scheme considering carbon emission

Remanufacturing scheme design (RSD) is a necessary process to improve the utilization of waste resources and reduce the remanufacturing carbon emissions. However, the lack of a mapping between the RSD parameters and carbon emissions makes it challenging to generate remanufacturing schemes that satisfy the low-carbon target. This would lead to large-scale remanufacturing of carbon emissions resulting from unreasonable RSD solutions, depriving remanufacturing of its energy saving and emission reduction benefits. To this end, this paper proposes an integrated design method for remanufacturing scheme considering the carbon emission. For establishing the relationship between the RSD parameters and the carbon emissions, the axiomatic design (AD) matrix and sensitivity analysis method are applied to select the mapping model between RSD parameters and remanufacturing carbon emissions, which analyzed the correlation between the three levels of RSD-remanufacturing process-carbon emission. Then, a multi-constrained objective optimization model with the depreciation of remanufacturing carbon emissions as the optimization objective is established, which is set the RSD parameters as the optimization variables. Meanwhile, an improved bald eagle search optimization algorithm (IBES) is used to solve the optimization model, contributing to the rapid solution of an optimal remanufacturing scheme. Finally, the feasibility of the method is verified by the RSD of the used injection mold as an example. The results show that the optimized RSD parameters reduce carbon emissions by 35.48 % compared to the original ones, which also demonstrates that the RSD method can effectively reduce carbon emissions from the remanufacturing of used products, as well as greatly reduce carbon emissions from large-scale remanufacturing industries, and promote sustainable development of the global industry.

Topology Optimization of Thermal Insulators considering Thermal–Structural Multi-Objective Function

This study develops a two-dimensional design sensitivity analysis and topology optimization method for multi-objective problems considering both thermal insulation and structural stiffness. The objective function is defined by introducing a weight factor that combines the thermal total energy and compliance for the thermal and structural problems. The derived design sensitivities are validated using the finite difference sensitivities. The Pareto front for designing thermal insulation is obtained by varying the weight function of the objective function and size of the domain. This meant that the geometric feature of the thermal bottleneck outweighed the load-supporting structures as the relative importance of the thermal insulation increased. Compared with the existing designs, the obtained designs are both structurally reliable and thermally insulative in terms of quantitative performance measures.

A Fast Singular Boundary Method for the Acoustic Design Sensitivity Analysis of Arbitrary Two- and Three-Dimensional Structures

This paper proposes a fast meshless scheme for acoustic sensitivity analysis by using the Burton–Miller-type singular boundary method (BM-SBM) and recursive skeletonization factorization (RSF). The Burton–Miller formulation was adopted to circumvent the fictitious frequency that occurs in external acoustic analysis, and then the direct differentiation method was used to obtain the sensitivity of sound pressure to design variables. More importantly, RSF was employed to solve the resultant linear system obtained by the BM-SBM. RSF is a fast direct factorization technique based on multilevel matrix compression, which allows fast factorization and application of the inverse in solving dense matrices. Firstly, the BM-SBM is a boundary-type collocation method that is a straightforward and accurate scheme owing to the use of the fundamental solution. Secondly, the introduction of the fast solver can effectively reduce the requirement of computer memory and increase the calculation scale compared to the conventional BM-SBM. Three numerical examples including two- and three-dimensional geometries indicate the precision and efficiency of the proposed fast numerical technique for acoustic design sensitivity analysis associated with large-scale and complicated structures.

Singular boundary method for 2D and 3D acoustic design sensitivity analysis

In this paper, a novel Burton-Miller-type singular boundary method (BM-SBM) formulation is proposed for the acoustic design sensitivity analysis, with the help of the direct differentiation method. The Burton-Miller formulation is employed to overcome the fictitious frequency problem in the numerical solutions of exterior acoustic problems. The simple empirical formulas are used to estimate the origin intensity factors in the BM-SBM. Compared with the traditional methods, the present scheme is an accurate and semi-analytical approach with the merits of truly meshless, integration-free, mathematical simplicity, and easy-to-program. As a boundary-type method based on the fundamental solution, furthermore, the BM-SBM is straightforward for addressing the exterior Helmholtz problems encountered in the acoustic design sensitivity analysis. Numerical experiments, including 2D and 3D geometries with Neumann boundary conditions, indicate that the proposed methodology is an accurate, stable, and convergent numerical approach for the acoustic design sensitivity analysis.

Shape design optimization of dynamic crack propagation using peridynamics

Crack generation tends to be prevented in most of design problems but taking advantages of cracks with some flaw acceptability could provide better design margin in practical problems. To develop a crack control strategy, we propose a shape design optimization method using an adjoint shape sensitivity based on a peridynamics theory that does not require crack propagation path a priori and thus is advantageous for the problems in which the discontinuity initiates, propagates, and interacts dynamically. The adjoint design sensitivity analysis (DSA) method based on the peridynamics is developed for dynamic crack propagations in brittle materials, including the successive branching of cracks. For a performance measure of stretch between the designated points at final time, the design sensitivity with respect to the design parameters is derived to be used in the shape design optimization. Numerical examples are demonstrated to verify the accuracy of the developed design sensitivity to show excellent agreement with finite differencing for all the shape design variables. Also, the proposed DSA method works very well in the shape design optimization applications of creating or avoiding cracks in a specified region.

State-space based discretize-then-differentiate adjoint sensitivity method for transient responses of non-viscously damped systems

In this paper, a new design sensitivity analysis (DSA) method for transient responses of non-viscously damped systems is developed. The damping force of the non-viscously damped systems depends on the past history of motion and is represented by convolution integrals over suitable kernel functions. The equations of motion are first transformed into a state-space formulation and the time-history analysis of the responses relies on a modified precise integration method (MPIM). The residual equation in each discrete time step is achieved by a recurrence formula of the MPIM. The sensitivities of the transient responses are calculated with an adjoint variable method (AVM). In particular, a discretize-then-differentiate approach is adopted for the DSA. The accuracy, consistency, implementation effort and computational complexity are discussed. Two numerical examples are illustrated to show the performances of the proposed method compared with other two methods. The results indicate that, when calculating the sensitivities of the transient responses for non-viscously damped systems based on the MPIM, the order of differentiation and discretization has no obvious effect on the consistency. The proposed state-space based discretize-then-differentiate AVM is more efficient than other two methods.

Design sensitivity analysis with polynomial chaos for robust optimization

Polynomial chaos (PC) methods with Gauss-type quadrature formulae have been widely applied for robust design optimization. During the robust optimization, gradient-based optimization algorithms are commonly employed, where the sensitivities of the mean and variance of the output response with respect to design variables are calculated. For robust optimization with computationally expensive response functions, although the PC method can significantly reduce the computational cost, the direct application of the classical finite difference method for the analysis of the design sensitivity is impractical with a limited computational budget. Therefore, in this paper, a semi-analytical design sensitivity analysis method based on the PC method is proposed, in which the sensitivity is directly derived based on the Gauss-type quadrature formula without additional function evaluations. Comparative studies conducted on several mathematical examples and an aerodynamic robust optimization problem revealed that the proposed method can reduce the computational cost of robust optimization to a certain extent with comparable accuracy compared with the finite difference-based PC method.

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.

Topology design optimization of conductive thermal problems subject to design-dependent load using density gradients

In this article, a design sensitivity analysis method is developed for topology optimization of steady-state conductive thermal problems subject to design-dependent thermal loads using density gradients–based boundary detection. In a real physical heat conduction problem, a design-dependent thermal load through the convective boundary is an important factor; however, it is not easy to impose boundary conditions in heat conduction problem due to the difficulties of exact boundary representation. Here, the detection of convection boundaries is available by the density distinction between void and materials, which is similar to a gradient operator in image processing. Applying density gradients and Dirac delta function of densities, we impose a design-dependent load effect on thermal analysis and perform the design sensitivity analysis for topology optimization. At each optimization process, it is noted that the results of thermal analysis and the design sensitivities are influenced by the design-dependent thermal load. Through demonstrative numerical examples, the proposed approach using density gradients is proven to work yielding meaningful optimization results. The developed approach is applicable to the plant engineering and nuclear industries where thermo-mechanical coupling occurs inevitably by the thermal convection.

Isogeometric configuration design sensitivity analysis of geometrically exact shear-deformable beam structures

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method.

Design sensitivity analysis for transient response of non-viscously damped systems based on direct differentiate method

Abstract Calculation of the first and second derivatives of transient response with respect to design variables is a prerequisite when gradient-based methods are adopted for optimization design in time-domain. In this paper, a design sensitivity analysis (DSA) method for calculating the first and second derivatives of the transient response for non-viscously damped systems is developed. The assumed damping forces depend on the past history of motion via convolution integrals over some kernel functions. The direct differentiate method (DDM) is selected to derive the DSA method. By introducing a generalized damping model in expression of fraction formula, the equations of motion of the non-viscously damped system are transformed into a state-space form without the convolution integral terms. Then, the first and second derivatives of the transient response are formulated based on a modified precise integration method using the DDM. The numerical stability, accuracy and implementation effort of the DDM are discussed. Two numerical examples are comparatively demonstrated using the DDM, the discretize-then-differentiate adjoint variable method (AVM) and the differentiate-then-discretize AVM. The results indicate that, by considering all the computational considerations, the proposed DDM is more suitable than the other two methods for the sensitivity analysis of transient response for non-viscously damped systems. Besides, it is also the only existing method to capture the second-order derivatives of transient response for the non-viscously damped systems.

Optimal determination of force field parameters for reduced molecular dynamics model

Using a gradient-based optimization method, the time-consuming atomistic model of substrate is replaced by computationally efficient Lennard-Jones (L-J) potential walls whose parameters are determined to appropriately represent the interactions between the nanoparticles and the substrate. To obtain the required design sensitivity with respect to design variables for the constant temperature molecular dynamics (MD) simulations that use the Nosé–Hoover thermostat, the finite difference method is impractical due to the huge amount of computational costs. Thus, we developed an adjoint design sensitivity analysis (DSA) method that is efficient for the system of many design variables. In numerical examples, we replace the complicated and time-consuming silicate structure to a multiple layer model of L-J potential wall, through the design optimization that includes the design variables of ϵ, σ, and the positions of each layer. The objective is to minimize the squared difference of time averaged performance between the full and the reduced models during the whole time span. The proposed method could lead to a significant reduction of computational costs, together with comparable outcomes from MD simulations.

Isogeometric configuration design optimization of shape memory polymer curved beam structures for extremal negative Poisson’s ratio

Using a continuum-based design sensitivity analysis (DSA) method, a configuration design optimization method is developed for curved Kirchhoff beams with shape memory polymers (SMP), from which we systematically synthesize lattice structures achieving target negative Poisson’s ratio. A SMP phenomenological constitutive model for small strains is utilized. A Jaumann strain, based on the geometrically exact beam theory, is additively decomposed into elastic, stored, and thermal parts. Non-homogeneous displacement boundary conditions are employed to impose mechanical loadings. At each equilibrium configuration, an additional nonlinear analysis is performed to calculate the Poisson’s ratio and its design sensitivity of the SMP material. The design objectives are twofold: for purely elastic materials, lattice structures are designed to achieve prescribed Poisson’s ratios under finite compressive deformations. Also, SMP-based lattice structures are synthesized to possess target Poisson’s ratios in specified temperature ranges. The analytical design sensitivity of the Poisson’s ratio is verified through comparison with finite difference sensitivity. Several configuration design optimization examples are demonstrated.

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.

A comparative study of design sensitivity analysis based on adjoint variable method for transient response of non-viscously damped systems

Abstract In this paper, design sensitivity analysis methods for the transient response of non-viscously damped systems are considered. The damping forces of the non-viscously damped systems depend on the past history of motion via convolution integrals over some suitable kernel functions. The adjoint variable method (AVM) is adopted to develop the design sensitivity analysis. Two numerical solution schemes, namely the discretize-then-differentiate method and the differentiate-then-discretize method, are introduced to complete the AVM for the sensitivity analysis of non-viscously damped systems. The discretize-then-differentiate AVM discretizes the equations of motion based on the Newmark- β implicit integration method first and then differentiates the discrete equations. On the contrary, the differentiate-then-discretize AVM firstly differentiates the equations of motion after transforming it into a state-space form, and then discretizes the equations based on a modified precise integration method (PIM). The numerical accuracy, efficiency, consistency and implementation effort are discussed and compared. Two numerical examples are presented to show the effectiveness of both methods. The results indicate that, by considering both computational accuracy and efficiency, the PIM based differentiate-then-discretize method is more suitable than the Newmark- β based discretize-then-differentiate method for the sensitivity analysis of transient responses for non-viscously damped systems.

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.

Isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using Jaumann strain formulation

Using an isogeometric approach, a continuum-based configuration design sensitivity analysis (DSA) method is developed for curved Kirchhoff beams with multi-patch junctions. Under the total Lagrangian formulation, large deformations considering the initial curvature of curved beams are described by geometrically exact beam theory (GEBT) and Jaumann strain formulation. In the isogeometric approach, the higher order continuity and the exact description of initial geometry are naturally embedded using NURBS basis functions. In multi-patch models, C0-continuity of physical displacement or C1-continuity of displacement component at junction is weakly imposed using the Lagrange multiplier method. The superior accuracy of isogeometric analysis (IGA) is verified through the comparison with the results of finite element analysis (FEA) using cubic Hermite interpolation. In the DSA, a material derivative is utilized and the kinematical description of GEBT is consistently employed to express orientation design variations. Contrary to the IGA-based DSA, the Hermite basis function explicitly depends on design in the FEA-based DSA due to its element length parameter. Moreover, since the design velocity field is approximated using the nodal velocity imposed at nodal tangential vector, the amount of design perturbations should be very small to obtain precise design sensitivity.

Isogeometric shape design sensitivity analysis of elasticity problems using boundary integral equations

Using boundary integral equations and isogeometric approach, a shape design sensitivity analysis (DSA) method is developed for two dimensional elastic structures. In the isogeometric approach, NURBS basis functions in CAD systems are directly utilized in response analysis, which enables a seamless incorporation of exact geometry and higher continuity into computational framework. To enhance the accuracy of shape design sensitivity, the CAD-based higher-order geometric information such as curvature, normal, and tangential vector is exactly embedded in the sensitivity expressions. In boundary integral formulation, shape design velocity field is decomposed into normal and tangential components, which significantly affect the accuracy of shape design sensitivity. Also, the proposed boundary-based method does not require the tedious design parameterization of internal domain. Through the numerical examples, the developed shape DSA method turns out to be more accurate than conventional finite element based one.