Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations

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.

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.

Two of the most important requirements when using gradient-based optimization for fluid–structure interaction problems are efficiency and accuracy in calculating the sensitivities. As a result, analytical continuum sensitivity formulations are finding their place due to their lower cost and accuracy. However, the necessary computation of mesh sensitivities in shape optimization is becoming a bottleneck, especially when handling complex geometries. In this research, an algorithm based on continuum sensitivity analysis for structural shape design variables is developed. In the proposed method, regularized Heaviside functions are used to modify the properties of mesh cells. Therefore, mesh dependency is removed from the sensitivity equations. This method does not require moving or modifying the mesh to handle the shape changes. This method is applied to two different validation cases where continuum sensitivity equations are formulated and solved. Validation cases are selected as a structural and thermal problem to show the applicability of this method for different physics. The results demonstrate the efficiency of the proposed algorithm for design-optimization framework.

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.

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.

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.

The computational cost of reliability analysis and optimization of medium/large structural systems is prohibitively high. One of the reasons for this situation is the use of inefficient methods, such as finite differences, for design sensitivity analysis. A significant increase in the efficiency of reliability computations is expected from the integration of reliability methods with the discrete and continuum methods of design sensitivity analysis. So far, only the discrete method has been used in the so-called stochastic finite element method. Integration of reliability computations with the continuum method of design sensitivity analysis is proposed in this paper as a new, powerful approach to the probabilistic design of large structural systems. Basic formulations and solution methods of reliability analysis, design sensitivity analysis of reliability indices, reliability-based optimization, and multiobjective reliability-based optimization are reviewed with focus on the sensitivity informatio...

Problems of motion of a rigid body with a fixed point are one of the urgent problems in classical mechanics. A feature of this problem is that, despite the important results achieved by outstanding mathematicians in the last two centuries, there is still no complete solution. This paper obtains an analytical solution of the problem of motion of an axisymmetric rigid body with variable inertia moments in resistant environment described by the system of nonlinear differential equations of L. Euler, involving the partial discretization method for nonlinear differential equations, which was built by A. N. Tyurekhodjaev based on the theory of generalized functions. To such problems belong gyroscopic instruments, in particular, and especially gyroscopes.

A unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed. A domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed. For numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface. In shape design problems, nodal points of the finite element model move as the shape changes. A method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid. For numerical implementation of the continuum shape design sensitivity analysis and automatic regridding, an established finite element analysis code is used. To demonstrate the feasibility of the method developed, shape design optimization of a main engine bearing cap is carried out as an example.

A simple method for the prediction of the stiffness-controlled voltage response of a hydrophone configured for deep-sea operation is described. The approach utilizes previous work in this area, enlarged to include the effects of various degrees of decoupling of the acoustic field from the back face of the hydrophone. Two configurations of importance, the spherical and the cylindrical, have been analyzed, and the results presented graphically. For both cases, the equations of equilibrium and the equation of compatibility are solved and evaluated by the use of boundary conditions. These stress equations, thus generated, and the electromechanical constants are then related to yield the voltage/pressure sensitivity equations. The graphs show the effects of thickness to diameter, comparison of the air-backed to the pressure-equalized hydrophone, and the various degrees of decoupling of the acoustic field from the back face of the hydrophone. For all cases, both barium titanate and lead zirconate titanate are considered.

In this paper, continuum-based sizing and configuration design sensitivity analysis (DSA) and optimization methods using the established transient dynamic analysis code DYNA3D are developed. For transient dynamic analysis, the explicit central difference method is used. For the structural domain, the updated Lagrangian formulation is used since it is appropriate for the path dependent problem. The first-order variations of the kinetic energy form, strain energy form, load form, and kinematic and structural responses with respect to the sizing and configuration design variables are derived. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs is updated at each updated configuration. The direct differentiation method is used in this paper for DSA. Numerical implementation of sizing DSA, configuration DSA, and optimization of a vehicle frame and A-pillar models has been carried out using the nonlinear finite element analysis (FEA) code DYNA3D.

In this thesis we analyze and develop methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs). The methods combine a direct solution approach with model reduction via proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). The reduced-order models (ROMs) are used to approximate the high-dimensional dynamic systems originating from a spatial discretization of a PDE. However, when used in an optimization algorithm, conventional POD/DEIM ROMs often lack the ability to give adequate approximations of the gradient. We propose methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces. We distinguish two types of error between quantities evaluated with the high-dimensional model and its ROM approximation in dependency on the optimization variable q: The reconstruction error which is evaluated with the same q0 which is used constructing the ROM and the prediction error which assesses approximations at q with a ROM constructed at q0 different to q. The novel reconstruction results we present include estimates for solutions of the adjoint equation and the sensitivity equations as well as for the gradient of the objective function. Based on the estimates we explain how the POD and DEIM bases should be extended with either adjoint or sensitivity information. The enhanced ROMs allow control of the reconstruction error for the objective and its gradient up to machine precision. Moreover, we propose a POD prediction estimate for the objective of the optimization problem in a neighborhood of q where the ROM is constructed. In case of sensitivity-extended POD and DEIM bases we give an analogous result for solutions of the states. The derivative-extended ROMs are then used to develop adaptive algorithms for the solution of optimal control and parameter estimation problems which results in great runtime improvements for the optimization while ensuring high approximation quality of the solution of the original problem. For the parameter estimation case a novel a posteriori error estimate is proposed which assesses the quality of suboptimal solutions obtained with the ROM. A further fundamental contribution is a discussion of discretize-then-optimize (DTO) vs. optimize-then-discretize (OTD) approaches in the context of MOR for optimization. We analyze advantages and disadvantages of both approaches and discuss to which extent our methods exhibit properties of either strategy. We also give examples of representative optimization problems in which standard POD/DEIM ROMs show an inacceptable behavior and can be successfully solved by derivative-extended ROMs. We have further implemented the developed methods emphasizing an efficient realization which is important for the investigation of the MOR potential. We showcase the practical performance of the proposed algorithms and the superiority of derivative-extended over conventional ROMs on two academic and one industry-relevant application which exhibit a variety of challenges for the model reduction approach in optimization.

The one- and two-dimensional inviscid Euler equations are formulated in an integro-differential form for shape design sensitivity analysis and optimization. The principal tool employed to derive the performance derivative sensitivity equations is the variational method, which is a continuous alternative to the discrete sensitivity analysis. Along with the sensitivity equations, the co-state equations and their boundary conditions are derived. Thereafter, based on the modal analysis of Von Neumann theory, the stability limits of the co-state equations are investigated. The stability characteristics of the co-state equations are compared with those of the Euler equations. Finally, using the criteria from the stability analysis of the state and co-state equations, some results of shape design optimization and sensitivity analysis for both quasi one- and two-dimensional Euler equations are presented. Copyright © 1999 John Wiley & Sons, Ltd.

This article presents a general formulation of the continuous sensitivity equation method for turbulent heat transfer. The standard k– e model of turbulence with wall functions is used to model turbulence, while turbulent heat transfer is handled using an eddy conductivity. The formulation accounts for complex parameter dependence and is suitable for a wide range of problems. In addition to a detailed description of the formulation, we perform careful verification of the methodology on a problem possessing a closed-form solution. The methodology is then demonstrated by performing a detailed sensitivity analysis of the turbulent forced convection over a heated flat plate. Details include sensitivity of the Stanton number with respect to the closure coefficients of the k– e model and the parameters appearing in the momentum and thermal wall functions.