Adjoint Variable Method Research Articles

Reliability-based topology optimization of fractionally-damped structures under nonstationary random excitation

This contribution proposes an effective reliability-based topology optimization (RBTO) framework for the layout of viscoelastic dampers of energy-dissipating structures under nonstationary seismic excitation, in which the constitutive behavior of viscoelastic damper is described by the novel fractional derivative models. To formulate the RBTO problem, the installation number of fractional viscoelastic dampers is minimized under the constraints of dynamic reliability, and the design variables are chosen as the damper parameters and the existence variables of the potential fractional viscoelastic dampers. The explicit responses are first established for the fractionally-damped structure, and the explicit response sensitivities with respect to the design variables are further derived through the adjoint variable method (AVM). On this basis, an explicit time-domain method (ETDM) is developed for explicit formulation of the statistical moments of responses and the moment sensitivities of the fractionally-damped structure, which are further employed to analytically derive the first-passage dynamic reliabilities and the reliability sensitivities of the structure based on the classical level-crossing process theory. Finally, the RBTO problem for the layout of fractional viscoelastic dampers is solved with the gradient-based method of moving asymptotes (MMA), in which the existence variables of fractional viscoelastic dampers can be driven to binary solutions by the solid isotropic material with penalization (SIMP) technique. An engineering application involving a building frame structure installed with fractional viscoelastic dampers is presented to validate the feasibility of the proposed ETDM-based RBTO framework.

Double Optimization Design of the Formula Racing Car Frame Based on the Variable Density Method and the Joint Variable Method

In the Formula Student Racing Car, the frame is a fundamental component that supports the body. The frame’s anti-deformation ability will impact the four-wheel positioning parameters of the car, which subsequently affect the car’s stability. The quality of the frame directly determines the power and efficiency of the racing car, making the frame crucial to the overall race performance. Therefore, researching lightweight frame design is particularly important to enhance frame performance and reduce its total weight. In this paper, based on the variable density method, the global topology optimization of the frame is carried out to obtain the distribution of the frame material, realize the efficient utilization of the material, and improve the torsional stiffness of the frame. Compared with the previous local topology optimization, the global topology optimization involves a wider range, and the results are more accurate. Based on the adjoint variable method, the sensitivity analysis of the frame is carried out to obtain the influence level of each design variable. The size of the frame is optimized according to the variables with high influence levels. After optimization, the total mass of the frame is reduced by 12.8%; the performance in terms of maximum displacement and maximum stress is improved; and the lightweight design of the frame is realized as a whole.

A BEM broadband topology optimization strategy based on Taylor expansion and SOAR method—Application to 2D acoustic scattering problems

AbstractIn this article, an innovative method is proposed for broadband topology optimization of sound‐absorbing materials adhering to the surface of a sound barrier structure. Helmholtz equation for the acoustic problems is solved using the boundary element method, while sensitivities of the objective function with respect to a large number of design variables are calculated via a sensitivity analysis method originated from the adjoint variable method (AVM), and the optimal solution is obtained by the method of moving asymptotes (MMA). Since the traditional single‐frequency topology optimization is frequency dependent, that is, the optimal solution at a certain frequency may not be optimal at other frequencies, broadband topology optimization of sound‐absorbing materials is conducted in this work. To address the problem of tremendous computational cost due to repetitive calculations at each discrete frequency point during broadband optimization, Taylor expansion of the Hankel function is performed to decouple the frequency dependent and independent terms in the BEM equations. For large‐scale problems, a reduced‐order model that retains the essential structures and key properties of the original model is built using the second‐order Arnoldi (SOAR) method. The accuracy and feasibility of the proposed algorithms are finally validated via some two‐dimensional numerical examples.

Concurrent Topology Optimization of Multi-Scale Composite Structures Subjected to Dynamic Loads in the Time Domain

This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: “discretize-then-differentiate” and “differentiate-then-discretize” approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the “discretize-then-differentiate” AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the “differentiate-then-discretize” AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted.

Acoustic sensitivity analysis for 3D structure with constant cross-section using 2.5D singular boundary method

In this paper, a 2.5D singular boundary method (SBM) in conjunction with the direct differentiation method (DDM) and adjoint variable method (AVM) are formulated for the sensitivity analysis of 3D longitudinally invariant structures. The assumption of a constant cross-section in the longitude direction enables it possible to decouple the 3D system into 2D problems at every wavenumber, and thus makes it a 2.5D-problem. The 2D sensitivity problem is solved by a boundary-type method, SBM. The SBM solves a problem with a linear combination of the fundamental solutions with respect to boundary collocation points. The fundamental solution makes the SBM available for exterior problems without the artificial truncated boundary. The source singularity issue of fundamental solution is overcome by a simple analytical formula and its derivatives. Numerical experiments present the accuracy, efficiency and feasibility of the proposed methods, and indicate that the proposed methods can be considered as an effective alternative in 3D problems with a longitudinally invariant cross-section.

Microstructure design of porous viscoelastic composites with prescribed relaxation moduli by multi-material topology optimization

This paper presents a multi-material topology optimization method for designing the microstructures of porous viscoelastic composites with prescribed relaxation moduli. It is assumed that the composite materials are microscopically composed of periodic unit cells. The effective relaxation moduli of the composites are obtained using the homogenization method. The density-based topology optimization is utilized, and a related material interpolation scheme is introduced. The design objective is to determine an optimal material distribution that exhibits the prescribed modulus in both the glassy and rubbery regions. The design sensitivity is derived analytically using the adjoint variable method. A filtering scheme is adopted to impose a minimum length scale for each material phase. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. Various microstructures with prescribed relaxation moduli are presented and discussed. In addition, the bounds on the relaxation moduli of the optimized microstructures are compared with theoretical bounds.

Dynamic topology optimization of continuum structures considering moving mass excitations

The structures excited by the moving mass are widely existed in nature and engineering. The dynamic loads are related to time and space and greatly affect structural performances. However, the moving mass problems have not been considered in the topology optimization design of continuum structures. Therefore, this paper proposes a novel dynamic topology optimization method to design the continuum structures considering moving mass excitations. Firstly, the finite element formulation is established to solve the dynamic responses for the moving mass excitations by using the HHT-α method. Secondly, in the density-based framework, the dynamic topology optimization problem is formulated to minimize the mean strain energy constrained with the total volume, whose interpolation scheme is the rational approximation of material properties. Thirdly, the sensitivities of the optimization problem are derived by using the adjoint variable method in the discretize-then-differentiate scheme. Finally, numerical examples demonstrate the effectiveness of the proposed method and the influence of the moving mass excitations on the topological structures.

Efficient optimal design of mosaic-like PPDW devices for THz application using the adjoint variable method.

In the development of THz-wave circuits, parallel plate dielectric waveguide (PPDW) is a promising platform and recently some fundamental devices have been reported. In order to realize high performance PPDW devices, optimal design methods are crucial and as out-of-plane radiation does not occur in PPDW, mosaic-like optimal design appears to be appropriate for PPDW platform. In this paper, we present a novel and efficient mosaic-like design approach based on gradient method with adjoint variable method (AVM) to realize high performance PPDW devices for THz circuit applications. The design variables in the design of PPDW devices are efficiently optimized by utilizing the gradient method. The mosaic structure in the design region is expressed by using density method with an appropriate initial solution. In the optimization process, AVM is employed for an efficient sensitivity analysis. The usefulness of our mosaic-like design approach is confirmed by designing several PPDW devices, T-branch, three branch, mode splitting device, and THz bandpass filter. The proposed mosaic-like PPDW devices except bandpass filter achieved high transmission efficiencies at single frequency operation as well as at broadband operation. Furthermore, the designed THz bandpass filter achieved the desired flat top transmission property at the targeted frequency band.

Topology optimization of microwave frequency dividing multiplexers

We use material-distribution-based topology optimization to design a three-port frequency dividing multiplexer at microwave frequencies. That is, by placing a good electric conductor inside the design domain, we aim to design a passive device that splits the incoming signal’s frequencies into two frequency bands and transmits them to their respective output ports. The Helmholtz equation models the time-harmonic wave propagation problem. We solve the governing equation using the finite element method. The adjoint variable method provides the required gradients, and we solve the topology optimization problem using Svanberg’s MMA algorithm. In this study, we present a technique for modeling the distribution of a good electric conductor within the design domain. In addition, we derive a power balance expression, which aids in formulating a series of three objective functions. In each successive objective function, we add more information and evaluate its impact on the results. The results show that by selecting a suitable objective function, we achieve more than 93.7,% transmission for both the frequency bands. Moreover, the numerical experiments suggest that the optimization problem is self penalized and is sensitive to the initial design.

Multi-material topology optimization for additive manufacturing considering dimensional constraints

Recent advances in additive manufacturing have enabled the fabrication of structures in which members made of multiple materials are placed in appropriate positions. However, dimensional constraints must be considered in such a process because of the size limitations of the 3D printer. This paper proposes a topology optimization method that considers the dimensional constraints of each material component. An extended level-set method is used to represent multiple material phases in the structural design. Constraint functions are formulated to control the size of each component. The weighted covariance matrix is used for the bounding box constraint. The sensitivity is based on the topological derivative and adjoint variable method. The proposed method was applied to solving the minimum compliance problem for two- and three-dimensional numerical examples to demonstrate its effectiveness and its potential contribution to enriching designs in additive manufacturing.

Adjoint variable method for transient nonlinear electroquasistatic problems

Many optimization problems in electrical engineering consider a large number of design parameters. A sensitivity analysis identifies the design parameters with the strongest influence on the problem of interest. This paper introduces the adjoint variable method as an efficient approach to study sensitivities of nonlinear electroquasistatic problems in time domain. In contrast to the more common direct sensitivity method, the adjoint variable method has a computational cost nearly independent of the number of parameters. The method is applied to study the sensitivity of the field grading material parameters on the performance of a 320 kV cable joint specimen, which is modeled as a finite element nonlinear transient electroquasistatic problem. Special attention is paid to the treatment of quantities of interest, which are evaluated at specific points in time or space. It is shown that the method is a valuable tool to study this strongly nonlinear and highly transient technical example.

Topology optimization for dynamic flexible multibody systems using the Flexible Natural Coordinates Formulation

This work proposes a fully-coupled topology optimization methodology to optimize the structural components of flexible multibody systems. It employs the Adjoint Variable Method (AVM) and the Flexible Natural Coordinates Formulation (FNCF) to achieve constant Jacobians in the reduced mass, stiffness, and damping matrix when computing the sensitivity information. Although a fully-coupled approach is inherently computationally more expensive than a method based on equivalent static load, the use of FNCF can reduce the computational time of a fully-coupled approach by only computing these terms once per design iteration, as opposed to for each timestep. The methodology is demonstrated through two optimization cases on a slider-crank multibody system and compared with the commonly used equivalent static load approach. The second case showcases the inter-component and system-level optimization capabilities of the fully-coupled approach by minimizing the deformation energy of two flexible bodies while only optimizing one body’s structural design.

An Efficient Decoupled Reliability-Based Topology Optimization Method Based on a Performance Shift Strategy

AbstractIn this paper, a new efficient reliability-based topology optimization (RBTO) method is proposed for structures, in which the double-loop optimization is equivalently decoupled into a sequential process based on a performance shift strategy. First, a volume minimization RBTO formulation is built for structures considering displacement or compliance reliability constraints. Second, an efficient decoupling scheme is proposed to turn the double-loop RBTO into a series of deterministic topology optimization and reliability analysis. For the reliability analysis, the reliability probability is calculated based on the probability distribution function of the performance function, and the probability distribution function can be solved by the maximum entropy method based on the raw moments calculated by the multiplicative dimension reduction method. A performance shift strategy is then applied to build an equivalence between probabilistic constraint and deterministic constraint to formulate the deterministic topology optimization. Thirdly, the adjoint variable method is applied to obtain the sensitivity information for topological design variables, and a gradient-based optimization algorithm is used to update the design variables. Finally, four typical numerical examples are used to verify the advantage of the proposed method. Compared with the traditional sequential optimization and reliability assessment (SORA) method, the proposed RBTO method results in a design with a 5.4% smaller volume value to reach the same reliability index requirement for the cantilever beam, and the function calls are 354 times less than that of the SORA method.

Function expansion based topology optimization of NRD guide device using hybrid method of harmony search and gradient method

In recent years, optimization technique has attracted a lot of attention to develop high performance electromagnetic devices and various kinds of design strategies have been reported. Among them, topology optimization method has potential to develop innovative devices. In this paper, the function expansion based topology optimization method, which has been developed for design of photonic devices, is extended to be applied to design of nonradiative dielectric waveguide (NRD guide) devices. In the optimal design of NRD guide devices, solution search sometimes tends to fall into local minimum compared with the design of photonic devices. Therefore, we develop a novel hybrid optimization method combining harmony search which is one of the evolutionary algorithm and gradient method using adjoint variable method for robust and efficient design of NRD guide devices.

Shape and topology optimization method for fiber placement design of CFRP plate and shell structures

In this study, we present a novel non-parametric shape-topology optimization method for fiber placement and orientation design, aiming at controlling the stiffness of CFRP (Carbon Fiber Reinforced Plastics) shell structures. The sum of squared error norm for achieving the target displacements on the arbitrarily specified position is minimized under the length constraints as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the density gradient for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The solid isotropic material with penalization (SIMP) method is employed for topology optimization. The gradient functions derived are applied to the H1 gradient method for fibers to determine the optimal shape and topology. With the present method, the optimal fiber placement and orientation can be obtained without any parameterization. Numerical examples are solved to show the validity of the proposed method and the results are discussed.

Design of constrained-layer damping on plates to sound radiation based on isogeometric analysis and non-negative intensity

Non-negative intensity (NNI) is a quantity which avoids near-field cancellation effects in sound intensity and provides direct visualization of the surface contributions to the radiated sound power. Hence, minimizing the integration of Non-negative intensity on predefined surfaces is implemented to be the design objective of topology optimization for the constrained-layer damping design on plates in this work. Non-negative intensity can be easily computed based on the radiation modes and the particle velocity on the surfaces of interest. Regarding the radiation modes, an eigenvalue analysis for the acoustic impedance matrix is required. After evaluating the objective function, the gradients of the objective function are computed using an adjoint variable method (AVM). These gradients enable the optimization to be solved by the method of moving asymptotes (MMA). Finally, some numerical examples are presented to validate the proposed optimization approach. Numerical results show that the corner radiation properties of the plate can be suppressed by the optimization, minimizing the integration of Non-negative intensity.

Model updating and damage detection of jacket type platform using explicit and exact time domain sensitivity equation

In this paper, a new sensitivity equation is formulated for structural parameter estimation based on the time-history response. There are three methods for calculating the structural response variation with respect to changes in structural parameters in the time domain: Finite Difference Method (FDM), Direct Differentiation Method (DDM), and Adjoint variable Method (AVM). These methods are gradient-based, and they cannot compute the actual sensitivity. The proposed sensitivity equation explicitly calculates the sensitivity (closed-form sensitivity) in an exact manner and provides a linear relationship between the time history response and the variation of structural parameters. The efficiency and accuracy of the proposed method are examined for the finite element model updating of a jacket-type platform. Moreover, the effectiveness of the proposed method is evaluated in the presence of measurement and environmental errors. The performance of the proposed sensitivity equation is compared with the mentioned direct differentiation method.

Topology Optimization of Sound-Absorbing Materials for Two-Dimensional Acoustic Problems Using Isogeometric Boundary Element Method

In this work, an acoustic topology optimization method for structural surface design covered by porous materials is proposed. The analysis of acoustic problems is performed using the isogeometric boundary element method. Taking the element density of porous materials as the design variable, the volume of porous materials as the constraint, and the minimum sound pressure or maximum scattered sound power as the design goal, the topology optimization is carried out by solid isotropic material with penalization (SIMP) method. To get a limpid 0–1 distribution, a smoothing Heaviside-like function is proposed. To obtain the gradient value of the objective function, a sensitivity analysis method based on the adjoint variable method (AVM) is proposed. To find the optimal solution, the optimization problems are solved by the method of moving asymptotes (MMA) based on gradient information. Numerical examples verify the effectiveness of the proposed topology optimization method in the optimization process of two-dimensional acoustic problems. Furthermore, the optimal distribution of sound-absorbing materials is highly frequency-dependent and usually needs to be performed within a frequency band.

Numerical studies for defect topology identification based on the adjoint variable and the finite element methods

This paper describes the visualization of defect regions in a structure via hammering test that employs the level set type topology optimization based on the concept of the phase-field method. In this study, the residual between the calculated displacement and the observed displacement is defined as the performance function, with the identification problem of defect regions being formulated based on the adjoint variable and finite element methods. Numerical experiments were performed to investigate the effect of the regularization parameter, the number of observation point, the evaluation equation of convergence and the application of the weighted gradient. Consequently, it was found that the number, position and size of defect can be adjusted using the regularization parameter τ. In addition, it was seen that if the number of observation points and the evaluation equation in the iterative computation of topology update are appropriately given, the identified defect topology is close to the exact defect topology. Furthermore, it was found that defect topologies can be more accurately identified by applying weighted sensitivities.

Shape optimization method for strength design problem of microstructures in a multiscale structure

AbstractIn this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed: minimization of maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier‐Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.