Shot peening simulation using discrete and finite element methods

Modeling shot peening process is very complex as it involves the interaction of metallic surfaces with a large number of shots of very small diameter. Conventionally such problems are solved using the finite element software (such as ABAQUS) to predict the stresses and strains. However, the number of shots involved and the number of elements required in a real-life components for a 100% coverage that lasts a considerable duration of peening make such an approach impracticable. Ideally, a method that is suitable for obtaining residual compressive stresses (RCS) and the amount of plastic deformations with the least computational effort seems a dire need.In this paper, an attempt has been made to address this issue by using the discrete element method (DEM) in combination with the finite element method (FEM) to obtain reasonably accurate predictions of the residual stresses and plastic strains. In the proposed approach, the spatial information of force versus time from the DEM simulation is utilized in the FE Model to solve the shot peening problem as a transient problem. The results show that the RCS distribution obtained closely matches with that of the computationally intensive direct FEM simulation. It has also been established, in this paper, that this method works well even in the situations where the robust unit cell approaches are found to be difficult to handle.     

Sequential approximate multi-objective optimization using radial basis function network

In industrial design optimization, objectives and constraints are generally given as implicit form of the design variables, and are evaluated through computationally intensive numerical simulation. Under this situation, response surface methodology is one of helpful approaches to design optimization. One of these approaches, known as sequential approximate optimization (SAO), has gained its popularity in recent years. In SAO, the sampling strategy for obtaining a highly accurate global minimum remains a critical issue. In this paper, we propose a new sampling strategy using sequential approximate multi-objective optimization (SAMOO) in radial basis function (RBF) network. To identify a part of the pareto-optimal solutions with a small number of function evaluations, our proposed sampling strategy consists of three phases: (1) a pareto-optimal solution of the response surfaces is taken as a new sampling point; (2) new points are added in and around the unexplored region; and (3) other parts of the pareto-optimal solutions are identified using a new function called the pareto-fitness function. The optimal solution of this pareto-fitness function is then taken as a new sampling point. The upshot of this approach is that phases (2) and (3) add sampling points without solving the multi-objective optimization problem. The detailed procedure to construct the pareto-fitness function with the RBF network is described. Through numerical examples, the validity of the proposed sampling strategy is discussed.     

Sequential robust optimization of a V-bending process using numerical simulations

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.     

Sequential approximate optimization using dual subproblems based on incomplete series expansions

In this paper, dual formulations for nonlinear multipoint approximations with diagonal approximate Hessian matrices are proposed; these approximations for example derive from the incomplete series expansion (ISE) proposed previously. A salient feature of the ISE is that it may be used to formulate strictly convex and separable (recast) primal approximate subproblems for use in sequential approximate optimization (SAO). In turn, this allows for the formulation of highly efficient dual formulations, and different combinations of direct, reciprocal, and exponential intervening variables for the objective and the constraint functions may be used. Two frequently encountered problems in structural optimization, namely the weight minimization problem with sizing design variables and the minimum compliance topology optimization problem, are degenerate cases of the formulations we present. Computational experiments confirm the efficiency of our proposed methodology; to this end, comparative results for the method of moving asymptotes (MMA) are presented. Based on the paper entitled “Duality in Convex Nonlinear Multipoint Approximations with Diagonal Approximate Hessian Matrices Deriving from Incomplete Series Expansions,” presented at the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, VA, USA, September 2006, paper no. AIAA-2006-7090.     

Normalized explicit finite element approximate inverse preconditioning

Normalized explicit approximate inverse matrix techniques for computing explicitly various families of normalized approximate inverses based on normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference and finite element discretization of partial differential equations are presented. Normalized explicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear systems. Theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Topology optimization using the p-version of the finite element method

A multiresolution topology optimization approach is proposed using the p-version finite element method (p-version FEM). Traditional topology optimization, where a density design variable is assigned to each element, is suitable for low-order h-version FEM. However, it cannot take advantage of the higher accuracy of higher-order p-version FEM analysis for generating results with higher resolution. In contrast, the proposed method separates density variables and finite elements so that the resolution of the density field, which defines the structure, can be higher than that of the finite element mesh. Thus, the method can take full advantage of the higher accuracy of p-version FEM.     

Maintaining large time steps in explicit finite element simulations using shape matching

We present a novel hybrid method to allow large time steps in explicit integrations for the simulation of deformable objects. In explicit integration schemes, the time step is typically limited by the size and the shape of the discretization elements as well as by the material parameters. We propose a two-step strategy to enable large time steps for meshes with elements potentially destabilizing the integration. First, the necessary time step for a stable computation is identified per element using modal analysis. This allows determining which elements have to be handled specially given a desired simulation time step. The identified critical elements are treated by a geometric deformation model, while the remaining ones are simulated with a standard deformation model (in our case, a corotational linear Finite Element Method). In order to achieve a valid deformation behavior, we propose a strategy to determine appropriate parameters for the geometric model. Our hybrid method allows taking much larger time steps than using an explicit Finite Element Method alone. The total computational costs per second are significantly lowered. The proposed scheme is especially useful for simulations requiring interactive mesh updates, such as for instance cutting in surgical simulations.     

Efficient structure topology optimization by using the multiscale finite element method

Efficient SIMP and level set based topology optimization schemes are proposed based on the computation framework of the multiscale finite element method (MsFEM). In the proposed optimization schemes, the equilibrium equations are solved on a coarse-scale mesh and the design variables are updated on a fine-scale mesh. To describe more complex deformation, a multi-node coarse element is also presented in the MsFEM computation. In the MsFEM, a multiscale shape function is constructed numerically and employed to obtain the equivalent stiffness matrix and load vector of the multi-node coarse element. In the optimization schemes with the MsFEM, the coarse elements are divided into two categories: homogeneous and heterogeneous. For the homogeneous coarse elements, their multiscale shape functions are constructed only once before the iterations. Since the material distribution is varying locally in most of the iterations, one only needs to reconstruct them of a small part of the coarse elements where the material distribution is changed by comparison with that in the previous iteration step. This will save lots of computational cost. In addition, due to the independence of each coarse element, the constructions of the multiscale shape functions could be easily proceeded in parallel. In this work, the computational accuracy and efficiency of this method is investigated in detail, as well as the speedup ratio and parallel efficiency when using multiple processors to construct the multiscale shape functions simultaneously. Furthermore, several 2D and 3D examples show the effectiveness and efficiency of the proposed optimization schemes based on the MsFEM analysis framework.     

Levelset based fluid topology optimization using the extended finite element method

This study focuses on finding the optimal layout of fluidic devices subjected to incompressible flow at low Reynolds numbers. The proposed approach uses a levelset method to describe the fluid-solid interface geometry. The flow field is modeled by the incompressible Navier?CStokes equations and discretized by the extended finite element method (XFEM). The no-slip condition along the fluid-solid interface is enforced via a stabilized Lagrange multiplier method. Unlike the commonly used porosity approach, the XFEM approach does not rely on a material interpolation scheme, which allows for more flexibility in formulating the design problems. Further, it mitigates shortcomings of the porosity approach, including spurious pressure diffusion through solid material, strong dependency of the accuracy of the boundary enforcement with respect to the model parameters which may affect the optimization results, and poor boundary resolution. Numerical studies verify that the proposed method is able to recover optimization results obtained with the porosity approach. Further, it is demonstrated that the XFEM approach yields physical results for problems that cannot be solved with the porosity approach.     

Sequential approximate reliability-based design optimization for structures with multimodal random variables

Structural and Multidisciplinary Optimization - For practical engineering design problems, random variables tend to follow multimodal probability distributions when working at multiple operating...     

Moving curved mesh adaptation for higher-order finite element simulations

Higher-order finite element method requires valid curved meshes in three-dimensional domains to achieve the solution accuracy. When applying adaptive higher-order finite elements in large-scale simulations, complexities that arise include moving the curved mesh adaptation along with the critical domains to achieve computational efficiency. This paper presents a procedure that combines Bézier mesh curving and size-driven mesh adaptation technologies to address those requirements. A moving mesh size field drives a curved mesh modification procedure to generate valid curved meshes that have been successfully analyzed by SLAC National Accelerator Laboratory researchers to simulate the short-range wakefields in particle accelerators. The analysis results for a 8-cavity cryomodule wakefield demonstrate that valid curvilinear meshes not only make the time-domain simulations more reliable, but also improve the computational efficiency up to 30%. The application of moving curved mesh adaptation to an accelerator cavity coupler shows a tenfold reduction in execution time and memory usage without loss in accuracy as compared to uniformly refined meshes.     

Identifying and reducing critical lag in finite element simulations

End-to-end lag time can undermine the effectiveness of interactive displays. Analyzing lag components for an FES indicated that simulation time is the critical component     

High quality real-time Image-to-Mesh conversion for finite element simulations

In this paper, we present a parallel Image-to-Mesh Conversion (I2M) algorithm with quality and fidelity guarantees achieved by dynamic point insertions and removals. Starting directly from an image, its implementation is capable of recovering the isosurface and meshing the volume with tetrahedra of good shape. Our tightly-coupled shared-memory parallel speculative execution paradigm employs carefully designed contention managers, load balancing, synchronization and optimizations schemes. These techniques are shown to boost not only the parallel but also the single-threaded efficiency of our code. Specifically, our single-threaded performance is faster than both CGAL and TetGen, the state of the art sequential open source meshing tools we are aware of. The effectiveness of our method is demonstrated on Blacklight, the Pittsburgh Supercomputing Center’s cache-coherent NUMA machine. We observe a more than 82% strong scaling efficiency for up to 64 cores, and a more than 82% weak scaling efficiency for up to 144 cores, reaching a rate of more than 14.3 million elements per second. This is the fastest 3D Delaunay mesh generation and refinement algorithm, to the best of our knowledge.     

Robust topology optimization using a posteriori error estimator for the finite element method

In our work, we consider the classical density-based approach to the topology optimization. We propose to modify the discretized cost functional using a posteriori error estimator for the finite element method. It can be regarded as a new technique to prevent checkerboards. It also provides higher regularity of solutions and robustness of results.     

Sequential kriging optimization using multiple-fidelity evaluations

When cost per evaluation on a system of interest is high, surrogate systems can provide cheaper but lower-fidelity information. In the proposed extension of the sequential kriging optimization method, surrogate systems are exploited to reduce the total evaluation cost. The method utilizes data on all systems to build a kriging metamodel that provides a global prediction of the objective function and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and a metal-forming process design problem via finite element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method.     

An approximate optimization strategy using refined hybrid metamodel

An approximate model called metamodel or surrogate model is a mathematical model that numerically approximates response of a system during an engineering simulation process or test. The introduction of a metamodel makes it possible to express response defined in the design problem as a simple mathematical function of design variables. A metamodel can be built with response surface method (RSM), kriging, neural network, radial basis function, and so on. Each method has its advantages and disadvantages. A combined metamodel called hybrid model, ensemble model, or multiple surrogates has been developed to maximize each metamodel's strength. The hybrid model of this research includes RSM and kriging. Besides, a strategy to refine the hybrid metamodel is implemented by reducing design space. In this process, information related to Hessian is utilized for an unconstrained optimization problem, on the contrary feasibility for a constrained optimization problem. This research presents a new hybrid metamodel-based optimization strategy called refined hybrid metamodel. Five mathematical test problems, two-bar design, spring design, and propeller shaft design problems are solved with the suggested method, verifying its usefulness. Most of the optimal results with the proposed method are closer to exact solutions with smaller function evaluations than existing methods.     

Solving linear finite element systems by normalized approximate matrix factorization semi-direct methods

New normalized Extended to the Limit sparse factorization procedures in algorithmic form are derived yielding direct and iterative methods for the solution of finite element or finite difference systems of irregular structure. The proposed factorization procedures are chosen as the basis to yield normalized systems on which the Conjugate Gradient and Chebychev methods are implicitly applied. The application of the derived normalized implicit semi-direct methods on a two-dimensional elliptic boundary-value problem is discussed and numerical results are given.     

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis

The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) allows us to design structures which satisfy economical and safety requirements. However, in practical applications, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak convergence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and optimization iterations. This approach leads to low numerical efficiency, which is disadvantageous for engineering applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to parallel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed methodology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).     

Two dimensional shape optimization using partial control and finite element method for compressible flows

The objective of this study is to determine the two dimensional shape of a body located in a compressible viscous flow, where the applied fluid force is minimized. The formulation to obtain the optimal shape is based on an optimal control theory. An optimal state is defined as a state, in which the performance function defined as the integration of the square sum of the applied fluid forces is minimized due to a reduction in the applied fluid forces. Compressible Navier–Stokes equations are treated as constraint equations. In other words, the body is considered to have a shape that minimizes the fluid forces under the constraint of the Navier–Stokes equations. The gradient of the performance function is computed using the adjoint variables. A weighted gradient method is used as the minimization algorithm. The volume of the body is assumed to be the same as that of the initial body. In the case of the algorithm used in this study, both the creation of a structured mesh around the surface of the body and the smoothing procedure are employed for the computation of gradient. In this study, a remeshing technique based on the structured mesh around the body changing its configuration in the iteration cycle is employed. For the correction to keep the volume constant, the surface coordinates are moved along the radial direction. For the discretization of both the state and adjoint equations, the efficient bubble function interpolation presented previously by the authors [18] is employed. The algorithm, which is known as the partial control algorithm, is applied to the numerical procedure to determine the movement of the coordinates. In the case of the gradient method, in order to avoid the convergence of the final shape to the local minimum shape, the new algorithm, which is called the partial control algorithm, is presented in this study. In numerical studies, the shape determination of a body in a uniform flow field is carried out in 2D domains. The initial shape of the body is assumed to be an elliptical cylinder. The shape is modified by minimizing the applied fluid forces. Finally, the desired shape of a body, whose performance function is reduced and converged to a constant value, is obtained. By carrying out a procedure that involves the use of the partial control algorithm, the desired shape of a body, whose performance function is reduced further, is obtained. Stable shape determination of a body in a compressible viscous flow is carried out by using the presented method. It is indicated that the optimal shape can be obtained by using the partial control algorithm.