Sensitivity Of Objective Function Research Articles

A novel interval dynamic topology optimization methodology of piezoelectric structures under reliable active control

In this paper, considering the influence of uncertainties in piezoelectric intelligent structures with active vibration control law, a robust topology optimization method to reduce the sensitivity of structural dynamic flexibility to uncertainty factors for piezoelectric intelligent structures is proposed. The effect of active control can be equivalent to the damping. Considering the influence of uncertainty, the structural dynamic response of the system will be within the interval range. Due to the monotonicity of the steady-state displacement response on uncertain parameters and the monotonicity of dynamic flexibility on uncertain parameters are both difficult to determine, a more accurate upper and lower limit is obtained using the collocation method. With the constraints of reliability and volume, the robustness index is defined based on the dynamic flexibility of the structure and defined as the objective function to establish the robust topology optimization model. In terms of optimization solution, the reliability index is transformed into the constraint translation distance. The sensitivity of constraint functions and objective functions to design variables is derived by employing the chain derivative method and the adjoint vector method. Finally, two examples are used to verify the effectiveness of the proposed method.

Concurrent topology optimization of multiscale piezoelectric actuators

As an important component of smart structural devices, piezoelectric composites have strong macroscopic electromechanical coupling properties which often depend on their microstructural geometries and material parameters. Therefore, optimizing the microstructural design is crucial for achieving the desired macroscale effective properties. To this end, this paper conducts a concurrent multiscale topology optimization (TO) to optimize the material distribution of piezoelectric actuators, aiming to maximize the electrical and mechanical energy transmission to meet specific engineering requirements. Firstly, an energy method is used to homogenize the microstructures and obtain the effective parameters of the piezoelectric material. Then, the piezoelectric material with penalization and polarization (PEMAP-P) approach in conjunction with the adjoint method is employed to calculate the sensitivity of objective and constraint functions. The sensitivities at both the macroscale and microscale are obtained by considering the interscale coupling effects. Finally, the concurrent bi-level iteration based on the optimality criteria (OC) or the method of moving asymptotes (MMA) is conducted. Through this research, we have illuminated the relations between the optimized performance of the actuator and the material volume fractions at each scale. We have also compared in detail the differences between two-scale and single-scale designs, as well as the differences between the simplified half-symmetric geometric model and the full geometric model. We find that a larger volume fraction of material either at the macroscale or microscale generates a larger transmitted displacement magnitude under the same applied electric field, and for a given total material, the transmitted maximum increases with increasing microstructural volume fraction.

Dynamic Topology Optimization of Constrained Layer Damping Structure Considering Non-Uniform Blast Load

The non-uniform distributed pressure of impact wave is usually simplified into concentrated or uniform load equivalently in the optimization design of constrained layer damping structure. However, for the thin-walled structure, it becomes necessary to regard the load as a non-uniform distribution. In this paper, a topology optimization approach is proposed considering the blast load with non-uniform distribution, aiming to unveil its impact on optimization layouts and dynamic responses. Initially, the smoothed particle hydrodynamics (SPH) algorithm is used to obtain the blast pressure what are extracted and integrated into the optimization model. Subsequently, the relative density is regarded as design variable. The construction of material penalty model and the topology optimization model are based on polynomial interpolation scheme (PIS). The sensitivity of objective function is deduced employing an improved adjoint variable method (AVM) to fit the load forms, and the Newmark-[Formula: see text] method is used to calculate the dynamic response. The optimization criterion (OC) is adopted to update the design variables. Finally, two numerical examples are used to exhibit the validity and accuracy of the presented methodology. The findings indicate that the distributed form, spread velocity, excited position and excited amplitude of the blast load all exert a notable influence on optimization results and dynamic response. These results underscore the valuable engineering application of this research and introduce a fresh perspective to the challenge of topology optimization under the blast case.

Dynamic-constrained multi-material topology optimization under heat flux and thermal–mechanical loading

In practice, the structures are often simultaneously related to thermal and mechanical loads, which can lead to stability failure. Improving the stability of the structures is needed to keep them safe. This study aims to optimize vibration capacity in a thermal environment by using an extended solid isotropic material with penalization (SIMP). The effects of thermal and mechanical loadings are considered simultaneously with dynamic constraints as a first effort that includes heat fluxes around boundaries. The incorporation of heat flux boundary conditions is explored in the multi-physics optimization instead of uniform heating using multi-materials. The results show that the distribution of materials changes significantly when thermal properties are changed. The adjoint method is used to analyze the sensitivity of objective functions, and the method of moving asymptotes (MMA) is employed to update new design variables. The effectiveness and reliability of the proposed method are verified by some examples.

Globaliosios sijyno optimizacijos uždaviniai

Important problem in civil engineering is obtaining optimal pile placement schemes for grillage-type foundations in order to reduce the number of piles and reactive forces arising in piles. The idealizations of problem and formulation for local and global optimization are discussed, highlighting the must for global optimization. The grillage is discretized using finite element method. Sensitivity of objective function is obtained analytically. The solution technology for a class of aforementioned problems is proposed based on the use of global optimization algorithms of package GAMS, which are implemented as ``black boxes”. Results of computational experiments are presented.

Multi-objective shape optimization of doubly offset serpentine diffuser using Adjoint method

Doubly offset serpentine diffusers are needed in the compact designs of modern stealth fighters with highly integrated propulsion systems. In addition, these doubly offset serpentine diffusers are also essential to provide reduced radar cross-section (RCS) due to the blockage of a complete line of sight. In this research, shape optimization of the doubly offset serpentine diffuser is achieved using a multi-objective approach to maximize the pressure recovery (PR) and minimize the flow swirl angle (α) and distortion coefficient (DC) at the aerodynamic interface plane (AIP). A computationally efficient Adjoint-based method was implemented for the optimization. For baseline geometry, an additional offset is added to NASA’s M2129 single serpentine duct to make it doubly offset serpentine diffusers. The freeform deformation technique was used to parametrize the baseline shape using 180 control points. A steady-state flow solution through the diffuser was achieved by solving Reynolds-Averaged Navier-Stokes (RANS) equations using a general-purpose computational analysis tool, i.e., ANSYS Fluent. The sensitivity of objective functions from the Adjoint solver was used to compute the required design change in the direction of minima or maxima of the objective functions using the steepest descent optimization technique. Subsequently, the calculated design changes were implemented through mesh Morpher. The Adjoint-based approach and freeform deformation resulted in a powerful and efficient optimization framework. The optimized shape showed a significant reduction in distortion coefficient and flow swirl angle, along with a slight increase in pressure recovery.

Concurrent topology optimization of multiscale structure under uncertain dynamic loads

This study proposes a robust concurrent topology optimization method with considering dynamic load uncertainty for the design of structures composed of periodic microstructures based on the bi-directional evolutionary structural optimization (BESO) method. The objective function is formulated as the summation of the mean and standard deviation of the structural dynamic compliance modulus. The constraints are imposed on the macrostructure and material microstructure volumes, respectively. The hybrid dimension reduction method and Gauss integral (HDRG) method is proposed to quantify and propagate load uncertainty to estimate the objective function. By the HDRG method, robust topology optimization with uncertainty modeled by probabilistic methods can be handed uniformly. To reduce the computational burden, a decoupled sensitivity analysis method is proposed to calculate the sensitivities of objective function with respect to the microstructure design variables. Five numerical examples are used to validate the effectiveness of the proposed robust concurrent topology optimization method and demonstrate the influence of load uncertainty on the design results. Results illustrate that the proposed methods can obtain the clear topologies of macro and micro structures, and the dynamic load uncertainty has a significant impact on the design results.

Topology optimization using the discrete element method. Part 1: Methodology, validation, and geometric nonlinearity

Structural Topology optimization is attracting increasing attention as a complement to additive manufacturing techniques. The optimization algorithms usually employ continuum-based Finite Element analyses, but some important materials and processes are better described by discrete models, for example granular materials, powder-based 3D printing, or structural collapse. To address these systems, we adapt the established framework of SIMP Topology optimization to address a system modelled with the Discrete Element Method. We consider a typical problem of stiffness maximization for which we define objective function and related sensitivity for the Discrete Element framework. The method is validated for simply supported beams discretized as interacting particles, whose predicted optimum solutions match those from a classical continuum-based algorithm. A parametric study then highlights the effects of mesh dependence and filtering. An advantage of the Discrete Element Method is that geometric nonlinearity is captured without additional complexity; this is illustrated when changing the beam supports from rollers to hinges, which indeed generates different optimum structures. The proposed Discrete Element Topology Optimization method enables future incorporation of nonlinear interactions, as well as discontinuous processes such as during fracture or collapse.

Multi-objective optimization of a small scale SCO2 turbine rotor system with a shaft cooler

The SCO2 turbine machines exchange energy through supercritical carbon dioxide. Their impeller has the features of high-temperature and −speed to enhance energy conversion efficiency, but the rotor needs to be cooled to be compatible with bearings and seals. The paper introduces a pivotal parameter optimization of a concentrating solar SCO2 turbine rotor and seeks to control the harmonic response amplitude while preserving the distance between the critical speed and the working speed. The optimization considers several parameters including bearing span, stiffness, effective mass and damping of the bearing hub, and gas film stiffness coefficients of the cooler. The optimization is accomplished using a multi-objective and −scale quantum harmonic oscillator algorithm (mMQHOA) that couples an information interaction algorithm and transfer matrix model. The application of information interaction accelerates the convergence speed of the objective functions. The verification results from the three-dimensional finite element (3D-FE) indicate that the non-dominant design reduces resonance amplitude of the disc by approximately 71.91%, while the critical frequency increases by about 34.33% in the direction away from the operating frequency, and imply a trade-off relationship between harmonic response amplitude and critical speed. It is further reveal that the increased gas film stiffness of cooler in the primary level interval (<1E6 N/m) has no significant effect on the harmonic response of the system. The optimization is based not only on the analysis of design parameters, but also focuses on the sensitivity of objective functions that can significantly affect dynamic performance. The models with a single variable of bearing span and film stiffness are investigated respectively, and then the sensitivity of the system response is analyzed. In addition, three different objective functions are proposed, with the purpose of constructing a universally applicable model that can be further used to optimize the analogous bearing rotor system.

Machine learning based topology optimization of fiber orientation for variable stiffness composite structures

AbstractThis study proposes a machine learning (ML) based approach for optimizing fiber orientations of variable stiffness carbon fiber reinforced plastic (CFRP) structures, where neural networks are developed to estimate the objective function and analytical sensitivities with respect to design variables as a substitute for finite element analysis (FEA). To reduce the number of training samples and improve the regression accuracy, an active learning strategy is implemented by successively supplying effective samples along with the suboptimal process. After proper training of neural networks, a quasi‐global search strategy can be applied by implementing a large number of initial designs as starting points in the optimization. In this article, a mathematical example is first presented to show the superiority of the active learning strategy. Then a benchmark design example of a CFRP plate is scrutinized to compare the proposed ML‐based with the conventional FEA‐based discrete material optimization (DMO) method. Finally, topology optimization of fiber orientations is performed for design of a CFRP engine hood, in which the structural performance generated from the proposed ML‐based approach achieves 12.62% improvement compared with that obtained from the conventional single‐initial design method. This article is anticipated to demonstrate a new alternative for design of fiber‐reinforced composite structures.

Topology optimization of constrained layer damping plates with frequency- and temperature-dependent viscoelastic core via parametric level set method

Frequency- and temperature-dependent properties of viscoelastic materials have a significant effect on vibration characteristics of constrained layer damping (CLD) structures. In this article, parametrized level set method (PLSM) is extended to solve the topology optimization problem of CLD plates, in which the viscoelastic material with frequency- and temperature-dependent characteristics is employed as damping core. The single/weighted modal loss factors, which are solved by an iterative method, are defined as objective functions, and topology optimization problem is formulated based on the finite element model for CLD plates which are described by combining parametric level set (PLS) functions. The sensitivities of objective functions with respect to expansion coefficients in PLS are calculated by the use of adjoint vector method. As the optimized results with complex constant shear modulus for viscoelastic core indicates, the proposed method is effective for topology optimization of CLD structures and insensitive to initial design. Numerical examples with the consideration of frequency- and temperature-dependent characteristics for “soft” and “hard” viscoelastic core are implemented. Through comparing the optimal results with different models for viscoelastic core, the influences of frequency- and temperature-dependent characteristics for viscoelastic core on natural frequencies, modal loss factors and optimized layouts are analyzed. Similarity indices are defined for distinguishing the difference among optimized layouts and initial design induced by modal strain energy contour. The optimal design dependence on layer thicknesses is further discussed with additional mass of CLD materials unchanged.

Robust Acoustic Objective Functions and Sensitivities in Adjoint-Based Design Optimizations

The multidisciplinary design of aircraft typically includes considerations of performance, weight, fuel burn, and noise, among other factors. Objective functions and constraints reflect these considerations to weigh the tradeoff between different designs. Higher-fidelity optimization exercises have used an adjoint approach to reach an optimal set of aerodynamic objective functions, such as increased lift and reduced drag. The adjoint approach significantly reduces the computational time required to find an optimal configuration when there are many design parameters and few objective functions and constraints. Including acoustics in an adjoint-based design optimization requires the sensitivity of acoustic objective functions with respect to design parameters. This paper presents an approach for defining the sensitivity of several acoustic objective functions, including time-integrated metrics, such as effective perceived noise level, and frequency-integrated metrics, such as overall sound-pressure level. A demonstration, validation, and details on the sensitivity analysis implementation in the second-generation Aircraft Noise Prediction Program are also shown.

Systemic Risk Driven Portfolio Selection

We consider an investor who trades off tail risk and expected growth of the investment. We measure tail risk through portfolio's expected losses conditioned on the occurrence of a systemic event: financial market loss being exactly at, or at least at, its VaR level and investor's portfolio losses being above their CoVaR level. We decompose the solution to the investment problem in terms of the Markowitz mean--variance portfolio and an adjustment for systemic risk. We show that VaR and CoVaR confidence levels control, respectively, the relative sensitivity of the investor's objective function to portfolio--market correlation and portfolio variance. Our empirical analysis demonstrates that the investor attains higher risk-adjusted returns, compared to well-known benchmark portfolio criteria, during times of market downturn. Portfolios that perform best under adverse market conditions are less diversified and invest on a few stocks which have low correlation with the market.

Vibration Isolation of Extended Ultra-High Acceleration Macro–Micro Motion Platform Considering Floating Stator Stage

It is a limitation for rapid development of microelectronics manufacturing industry to hardly overcome the acceleration limitation of macro–micro motion platform. The paper presents an extended ultra-high acceleration macro–micro motion platform to investigate breakthrough of acceleration limitation with driving modes “macro + micro + macro” (MMM). In the proposed platform, the number of the pair of VCMs was defined as n. Under ultra-high acceleration, floating stator stage is suggested to isolate the vibration and obtain superior performance of the platform. Its theoretical analyses including natural frequency analysis, transient response analysis and frequency response analysis are performed to verify vibration isolation of floating stator stage applied in the extended ultra-high acceleration macro–micro motion platform. The change trends and sensitivities of related objective functions incorporating vibration transmissibility, settling time and the maximum stroke of stator’s motion are explored with their related parameters, and their multi-objective optimization designs are carried out to achieve superior performances of different extended platforms. Moreover, one case is performed when n = 1. Its theoretical analyses, change trends and sensitivities are investigated. The superior performance of the platform is obtained to realize vibration isolation by multi-objective optimization. And its experiment of vibration isolation is also testified finally. The results provide a theoretical and technical basis on microelectronics manufacturing equipment upgrading and manufacturing rapid development.

A modified leaky ReLU scheme (MLRS) for topology optimization with multiple materials

Based on the leaky rectified linear unit (ReLU) function, the paper proposes a modified leaky ReLU scheme (MLRS) for topology optimization with multiple materials. A continuous function formed by sequence of modified leaky ReLU functions are used for the material property interpolation. No extra design variables will be introduced for whatever number of materials considered. Thus, the computational cost is independent to material types. Since the modified leaky ReLU function only consists of two linear functions, the sensitivity of objective and constraint functions is easy to derive. Besides, the sensitivity of MSRL stays a proper constant number even when the design variable is sufficiently small, which may help the convergence. The discontinuous points of derivatives can be jumped across smoothly so that stable transitions can be guaranteed from one type of material to another. An optimal criteria (OC) based solution method is derived from the KKT condition, and an interior point method based solution procedure is introduced for more complex cases. The proposed method is general and straightforward, and is compatible with the existing optimization algorithms. Numerical examples including structural and thermal topology optimization are used to verify the proposed method. The results demonstrate its efficiency and stability.

A gradient-based shape optimization scheme via isogeometric exact reanalysis

PurposeThis paper aims to propose a gradient-based shape optimization framework in which traditional time-consuming conversions between computer-aided design and computer-aided engineering and the mesh update procedure are avoided/eliminated. The scheme is general so that it can be used in all cases as a black box, no matter what the objective and/or design variables are, whilst the efficiency and accuracy are guaranteed.Design/methodology/approachThe authors integrated CAD and CAE by using isogeometric analysis (IGA), enabling the present methodology to be robust and accurate. To overcome the difficulty in evaluating the sensitivities of objective and/or constraint functions by analytic method in some cases, the authors adopt the finite difference method to calculate these sensitivities, thereby providing a universal approach. Moreover, to further eliminate the inefficiency caused by the finite difference method, the authors advance the exact reanalysis method, the indirect factorization updating (IFU), to exactly and efficiently calculate functions and their sensitivities, which guarantees its generality and efficiency at the same time.FindingsThe proposed isogeometric gradient-based shape optimization using our IFU approach is reliable and accurate, as well as general and efficient.Originality/valueThe authors proposed a gradient-based shape optimization framework in which they first integrate IGA and the proposed exact reanalysis method for applicability to structural response and sensitivity analysis.

Optimization of the porous material described by the Biot model

The paper is devoted to the shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. At the macroscopic level, the porous material is described by the Biot model defined in terms of the effective medium coefficients, involving the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. By virtue of the homogenization, these coefficients are computed using characteristic responses of the representative unit cell consisting of an elastic solid skeleton and a viscous pore fluid. For the purpose of optimization, the sensitivity analysis on the continuous level of the problem is derived. We provide sensitivities of objective functions constituted by the Biot model coefficients with respect to the underlying pore shape described by a B-spline box which embeds the whole representative cell. We consider material design problems in the framework of which the layout of a single representative cell is optimized. Then we propose a sequential linearization approach to the two-scale problem in which local microstructures are optimized with respect to macroscopic design criteria. Numerical experiments are reported which include stiffness maximization with constraints guaranteeing minimum required permeability, and vice versa. Issues of the design and anisotropy and the spline box parametrization are discussed. In order to avoid remeshing, a geometric regularization technique based on injectivity constraints is applied.

Development of Optimization Methodology for Short Range Air-to-Air Missile

This paper illustrates the development and the use of conceptual sizing tools integrated with optimization tools. The conceptual sizing tool is based on empirical and simple analytical calculation that simple and robust enough for wide range of inputs. The genetic algorithm optimization method then used to explore the global optimum for given variables. System response analysis is conducted to gain insight about the sensitivity of objective function because of design variables.

Multi-phase field topology optimization of polycrystalline microstructure for maximizing heat conductivity

The present study proposes multi-scale topology optimization for polycrystalline microstructures applying a multi-phase field method. The objective function is to maximize the heat compliance of macrostructure and the equality constraint is the material volume of constituents in an alloy consisting of two components with different heat conductivity. Two-scale steady-state heat conduction problem based on a homogenization method is conducted. The Allen-Cahn non-conserved time evolution equation with the additional volume constraint scheme is employed as the optimization strategy for updating the crystal configuration. In the time evolution equation, sensitivities of objective function with respect to phase-field variables are considered to relate topology optimization to the multi-phase field method. It is verified from a series of numerical examples that the proposed method has great potential for the development of material design underlying polycrystalline structure.

Multi-objective optimization of spherical roller bearings based on fatigue and wear using evolutionary algorithm

In Spherical Roller Bearings (SRBs) design the fatigue and wear lives are the most important factors. The fatigue life of bearing is connected to dynamic capacity (Cd) and wear life of bearing is linked with the elasto-hydrodynamic minimum film thickness (hmin). Multi-objective optimization (MOO) of SRBs has been considered in the present study. For SRBs optimization problem, two objectives (Cd and hmin), eight design variables, and twenty-two constraints have been considered. Bearing pitch diameter, roller diameter, number of rollers, effective roller length and the contact angle are five design geometrical variables and other three are constraint parameters. Objective functions have been optimized individually as well as simultaneously. Elitist Non Dominated Sorting Genetic Algorithm (NSGA-II) is used to solve a non-linear constrained optimization problem of the SRB design. A convergence methodology is performed to the bearing design for global optimum results. Results obtained from NSGA-II runs of MOO have been used to draw Pareto-optimal fronts (POFs). Optimum bearing dimensions are selected by considering the knee-point solution on the POFs. Results indicate that the dynamic capacity of optimized bearing got enhanced thus increase in life of the bearing. A sensitivity analysis is conducted to identify the sensitivity of objective functions with design variables. The sensitivity analysis plays an important part in deciding the tolerances, which can be provided to design variables for the manufacturing of optimized bearings. The results obtained from MOO problem have been compared with available literature and are found to be better.