Continuum Sensitivity Analysis for Structural Shape Design Variables Using Finite-Volume Method

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.

A Continuum Sensitivity Equation (CSE) method was developed for the uid ow problems to support computationally e cient optimization of aerodynamic design problems. The continuum sensitivity equations and sensitivity boundary conditions were derived using both local and total derivative form for potential ow around an airfoil. The nite element method was adopted for solving both of the original ow analysis problem and the sensitivity equations. In this paper, the pressure coe cient sensitivity with respect to the angle of attack was calculated for veri cation. Shape sensitivities with respect to airfoil shape parameters were also calculated, and the accuracy and e ciency between local form and total form CSE was compared.

In this work, we present a gradient optimization technique for optimizing deformation processes. The optimization is based on the continuum sensitivity method (CSM). CSM involves differentiation of the governing field equations of the direct problem (constitutive, contact and kinematic problems) with respect to the design variables and development of the weak forms for the corresponding continuum sensitivity equations. The present 3D developments involve a novel regularized approach to the contact sensitivity problem that addresses the non‐differentiability of the contact constraints. A relevant 3D die design problem is considered highlighting the features of the metal forming design simulator developed.

Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.

A hybrid global–local approach for optimization of injection gate locations in liquid composite molding process simulations

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g., hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, we derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. The approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.

Sensitivity analysis is essential for understanding how changes in design variables affect system performance. Numerical methods for calculating sensitivities, such as finite difference methods, are often easy to implement but can suffer from high computational costs and limited accuracy. Continuum Sensitivity Analysis (CSA) is an alternative approach for calculating analytical derivatives with respect to shape or value parameters. It is easy to implement and can be as accurate as conventional analytical sensitivity methods. By employing Spatial Gradient Reconstruction (SGR), continuous sensitivity equations can be solved in a nonintrusive manner. This method has been applied and validated on a wide range of problems. This work aims to extend the range of applicability of CSA to composite laminates. Composite materials may be characterized by several cross-coupling between loads and deformations, which leads to complex stress fields. A general formulation is introduced that applies to any plate-discretized Finite Element (FE) problem. The developed methodology utilizes the plate elements’ resultant forces and moments instead of the stresses to reconstruct spatial gradients and apply boundary conditions. Because of that, the method does not depend on the material type or lamination sequence and is also computationally inexpensive. It can be used indifferently for isotropic or composite materials, with any lamination sequence. This feature makes CSA attractive for classical FE models used in design optimization problems. This novelty extends the range applicability of CSA to any possible plate-based structural problem.

An integrated approach to the optimum design of control augmented structural systems is presented in which structural variables and control variables are changed simultaneously during the design process. Constraints are imposed on peak transient dynamic displacements and accelerations, static displacements, natural frequencies, and control system effort. Side constraints imposed on structural member sizes and control system thresholds and actuator output forces insure the generation of physically meaningful designs. Example problems are presented which bring out the benefits of simultaneous treatment of both the structural design variables and the control design variables.

Multi-objective free-form optimization for shape and thickness of shell structures with composite materials

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries

Numerical determination of boundary heat fluxes in an enclosure dynamically with natural convection through Fletcher–Reeves gradient method

This article proposes an electrode shape optimization strategy with state variable constraints. The continuum sensitivity formula is used for the electrostatic system to determine the extremum of objective functions with state variable constraints. We use the gradient projection method and a restoration process to treat geometric and state variable constraints. The geometric changes of the design parameter are determined by the velocity field calculated from the constrained sensitivity analysis and are expressed by the level set method. The application models are tested to demonstrate the feasibility and utility of the proposed method.

Compared with traditional metal-materials, fiber-reinforced composite materials have better performance in many aspects such as strength, stiffness, and fracture resistance. At present, fiber-reinforced composite materials have been widely used in automotive, aerospace, and other industrial fields. This paper proposes a topology optimization method for solving the fundamental frequency maximization of undamped free vibration of continuous fiber-reinforced composite structures. To achieve the simultaneous optimization of the structural topological configuration and the fiber angle. A dynamic topological optimization model is established with the permitted material usage as the constraint and the structure's first-order eigenvalue as the objective function. The model includes density design variables that characterizes the topological configuration of the structure and angular design variables that characterizes the fiber orientation. The analytical sensitivity formulas of the objective function of eigenvalue with respect to density design variables and angle design variables are derived in detail, and the method of moving asymptotes (MMA) is used to solve the optimization problem. Finally, three numerical examples are performed to verify the effectiveness of the proposed method, which includes a static optimization example with the stiffness maximization as the goal and two dynamic optimization examples with the first-order eigenvalue maximization as the goal. The results show that the proposed method can achieve a stable iterative history and fast convergence, and can effectively improve the structural frequency while achieving the integrated optimization of the structural topological configuration and the fiber angle.

Sensitivity analysis with the modified Heaviside function for the optimal layout design of multi-component systems