Efficient design sensitivity analysis of incompressible fluids using SPH projection method

Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

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.

Design Sensitivity Analysis (DSA) is a procedure for obtaining derivatives of design objectives, such as mass and strength with respect to some design parameters, such as size and shape. In the DSA methodology, we take an existing technique for computing objectives from parameters and systematically differentiate it so that the derivatives of the objectives can be computed directly, rather than employing some sort of differencing scheme to approximate derivatives. In this paper, we apply this technique to a non-linear finite element program, Radioss, and begin looking at the properties of the DSA methodology for crashworthiness simulations. It is shown that some major computational advantages can be gained by employing DSA for these kinds of problems. Besides cost savings, there are many cases where numerical differentiation results in erroneous derivatives due to numerical noise in crash simulations. It is shown that the DSA method avoids these difficulties, thus, making it quite important for use in support of design activities.

We present a fast first order explicit time integration scheme for solving parabolic problems in mechanics via standard numerical methods in space using irregular grids, such as unstructured finite element meshes, or grids containing elements or cells of very different sizes. The new scheme extends one of the explicit FIC-Time (EFT) integration methods derived by the authors in [23] that allow considerable larger time steps than the forward-Euler (FE) scheme. The new EFT scheme overcomes the limitations in the time step size of explicit time integration schemes for irregular grids containing large and small elements. A variable time step is used for eliminating the oscillations near Dirichlet boundaries when large time steps are used. The advantages of the new EFT scheme versus the FE scheme are shown in one-, two- and three-dimensional transient heat conduction problems using irregular finite element grids.

Explicit and implicit methods for design sensitivity analysis of nonlinear structures subjected to transient dynamic loads are described. The direct differentiation method is used to calculate the sensitivities. Elastic and elasto-plastic problems undergoing large deformation are treated in the formulation. The explicit method is quite straightforward to implement for analysis as well as design sensitivity analysis: knowing all the quantities at time t, one updates the quantities for the next time point without solution of any linear system of equations, or any iterations. For the implicit method, the effective stiffness matrix for the sensitivity equation is different from that for the analysis phase due to inclusion of large deformation in the formulation. A numerical procedure to integrate the equation is proposed which uses the symmetric effective stiffness matrix from the analysis phase. Although the sensitivity analysis problem is linear, the procedure requires iterations for sensitivity calculations at each time step, as for the analysis phase. The implicit method is implemented using the pre-conditioned conjugate gradient iterative solution scheme. Numerical results for sensitivities indicate the implicit method to be more accurate than the explicit method for transient elastoplastic problems. However, the explicit method is substantially more efficient than the implicit method.

The objective of this study is to examine two analytical methods for the design sensitivity analysis of optimal control problems and to develop the analyses necessary for the application of the methods to thermal systems. The two methods are the direct differentiation method and the adjoint variable method. Results for these methods are compared with those obtained using central finite difference approximations for design sensitivity analysis. The adjoint variable method has the potential to reduce significantly computational times for design sensitivity analysis while sacrificing only a small amount of accuracy. As the number of functional requiring design sensitivity analysis increases, however, the adjoint variable method becomes less attractive and the direct differentiation method becomes more attractive.

Three methods for design sensitivity such as numerical differentiation, analytical method and semianalytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semianalytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

The numerical simulation of incompressible viscous flows using primitive variable formulation requires that the pressure be obtained by solving the pressure Poisson equation (PPE). It is mentioned in the literature on incompressible flows [1,2] that when all the pressure boundary conditions are of Neumann type, then the pressure compatibility condition (PCC) has to be satisfied for guaranteeing the existence of solution. The main aim of this paper is to show that the PCC is not an important issue as has been claimed in the literature [1–3]. On the other hand, it will be shown that the crucial issue, in numerical simulation of incompressible viscous flows, is to ensure solenoidality of the velocity field accurately. We will further show that the accuracy of solenoidality depends on the choice of pressure boundary conditions. The demonstration is based on the computation for 2-D lid-driven cavity problem for Re=100 on 1292 uniform grid.

Implicit incompressible SPH (IISPH) solves a pressure Poisson equation (PPE). While the solution of the PPE provides pressure at fluid samples, the embedded boundary handling does not compute pressure at boundary samples. Instead, IISPH uses various approximations to remedy this deficiency. In this article, we illustrate the issues of these IISPH approximations. We particularly derive Pressure Boundaries, a novel boundary handling that overcomes previous IISPH issues by the computation of physically meaningful pressure values at boundary samples. This is basically achieved with an extended PPE. We provide a detailed description of the approach that focuses on additional technical challenges due to the incorporation of boundary samples into the PPE. We therefore use volume-centric SPH discretizations instead of typically used density-centric ones. We further analyze the properties of the proposed boundary handling and compare it to the previous IISPH boundary handling. In addition to the fact that the proposed boundary handling provides physically meaningful pressure and pressure gradients at boundary samples, we show further benefits, such as reduced pressure oscillations, improved solver convergence, and larger possible time steps. The memory footprint of fluid samples is reduced and performance gain factors of up to five compared to IISPH are presented.

Design sensitivity analysis and topology optimization of displacement–loaded non-linear structures

Design sensitivity analysis of dynamic response of nonviscously damped systems

The objective of this work is to develop and implement efficient numerical procedures for gradient based shape optimization of steady, strongly coupled fluid–structure interaction problems involving turbulent flow. The governing equations are the Reynoldsaveraged Navier-Stokes equations combined with a large displacement formulation for the structure. The eddy viscosity is computed using either algebraic turbulence models or the two-equation k − ω model. The solution for state is obtained using finite element residual formulations based on a consistent weak formulation of the fluid–structure interaction problem. Due to the possible large displacements of the structure, the fluid mesh is updated using a modified elastic analogy. The resulting nonlinear equations are solved using an approximate Newton method, making it possible to do design sensitivity analysis (DSA) by the direct differentiation method in a very efficient way where the resulting sensitivity equations are solved by an incremental iterative method. Gradient based shape optimization is illustrated for finding the shape of both an infinitely stiff and a flexible valve. Such examples of semi-analytic DSA and gradient based shape optimization of strongly coupled fluid-structure interaction problems involving turbulent flow modeled using two-equation turbulence models have, at least to the knowledge of the authors, not been presented before in the literature and thus represent novel results.

We present an efficient library-based numerical method for simulating the Hodgkin-Huxley (HH) neuronal networks. The key components in our numerical method involve (i) a pre-computed high resolution data library which contains typical neuronal trajectories (i.e., the time-courses of membrane potential and gating variables) during the interval of an action potential (spike), thus allowing us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations; (ii) an algorithm of spike-spike corrections within the groups of strongly coupled neurons to account for spike-spike interactions in a single large time step. By using the library method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity, such as average firing rate, interspike interval distribution, power spectra of voltage traces. Moreover, our large time steps using the library method can break the stability requirement of standard methods (such as Runge-Kutta (RK) methods) for the original dynamics. We compare our library-based method with RK methods, and find that our method can capture very well phase-locked, synchronous, and chaotic dynamics of HH neuronal networks. It is important to point out that, in essence, our library-based HH neuron solver can be viewed as a numerical reduction of the HH neuron to an integrate-and-fire (I&F) neuronal representation that does not sacrifice the gating dynamics (as normally done in the analytical reduction to an I&F neuron).

<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>

This study investigates the full penetration simulation of piles from the ground surface, focusing on frictional contact modeling without mesh distortion. To overcome issues related to mesh distortion and improve solution convergence, the Arbitrary Lagrangian–Eulerian (ALE) adaptive mesh technique was implemented within both explicit and implicit time integration schemes. The numerical model was validated against field experiments conducted at Bothkennar, Scotland, using the Imperial College instrumented displacement pile (ICP) in soft clay, where the soil behavior was effectively represented using the modified Cam-Clay model and the Mohr–Coulomb model. The primary objectives of this study are to evaluate the ALE method performance in handling mesh distortion; analyze the effects of soil–pile interface friction, pile dimensions, and various dilation angles on pile resistance; and compare the effectiveness of explicit and implicit time integration schemes in terms of stability, computational efficiency, and solution accuracy. The ALE method effectively modeled pile penetration in Bothkennar clay, validating the numerical model against field experiments. Comparative analysis revealed the explicit time integration method as more robust and computationally efficient, particularly for complex soil–pile interactions with higher friction coefficients.