On body shapes providing maximum depth of penetration

The problem of maximization of the depth of penetration of rigid impactor into semi-infinite solid media (concrete shield) is investigated analytically and numerically using two-stage model and experimental data of Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997). The shape of the axisymmetric rigid impactor has been taken as an unknown design variable. To solve the formulated optimization problem for nonadditive functional, we expressed the depth of penetration (DOP) under some isoperimetric constraints. We apply approaches based on analytical and qualitative variational methods and numerical optimization algorithm of global search. Basic attention for considered optimization problem was given to constraints on the mass of penetrated bodies, expressed by the volume in the case of penetrated solid body and by the surface area in the case of penetrated thin-walled rigid shell. As a result of performed investigation, based on two-term and three-term two stage models proposed by Forrestal et al. (Int J Impact Eng 15(4):396–405, 1994), Forrestal and Tzou (Int J Solids Struct 34(31–32):4127–4146, 1997) and effectively developed by Ben-Dor et al. (Comp Struct 56:243–248, 2002, Comput Struct 81(1):9–14, 2003a, Int J Solids Struct 40(17):4487–4500, 2003b, Mech Des Struct Mach 34(2): 139–156, 2006), we found analytical and numerical solutions and analyzed singularities of optimal forms.     

Cross-validation based single response adaptive design of experiments for Kriging metamodeling of deterministic computer simulations

A new approach for single response adaptive design of deterministic computer experiments is presented. The approach is called SFCVT, for Space-Filling Cross-Validation Tradeoff. SFCVT uses metamodeling to obtain an estimate of cross-validation errors, which are maximized subject to a constraint on space filling to determine sample points in the design space. The proposed method is compared, using a test suite of forty four numerical examples, with three DOE methods from the literature. The numerical test examples can be classified into symmetric and asymmetric functions. Symmetric examples refer to functions for which the extreme points are located symmetrically in the design space and asymmetric examples are those for which the extreme regions are not located in a symmetric fashion in the design space. Based upon the comparison results for the numerical examples, it is shown that SFCVT performs better than an existing adaptive and a non-adaptive DOE method for asymmetric multimodal functions with high nonlinearity near the boundary, and is comparable for symmetric multimodal functions and other test problems. The proposed approach is integrated with a multi-scale heat exchanger optimization tool to reduce the computational effort involved in the design of novel air-to-water heat exchangers. The resulting designs are shown to be significantly more compact than mainstream heat exchanger designs.     

Topology optimization of 2D continua for minimum compliance using parallel computing

Topology optimization is often used in the conceptual design stage as a preprocessing tool to obtain overall material distribution in the solution domain. The resulting topology is then used as an initial guess for shape optimization. It is always desirable to use fine computational grids to obtain high-resolution layouts that minimize the need for shape optimization and postprocessing (Bendsoe and Sigmund, Topology optimization theory, methods and applications. Springer, Berlin Heidelberg New York 2003), but this approach results in high computation cost and is prohibitive for large structures. In the present work, parallel computing in combination with domain decomposition is proposed to reduce the computation time of such problems. The power law approach is used as the material distribution method, and an optimality criteria-based optimizer is used for locating the optimum solution [Sigmund (2001)21:120–127; Rozvany and Olhoff, Topology optimization of structures and composites continua. Kluwer, Norwell 2000]. The equilibrium equations are solved using a preconditioned conjugate gradient algorithm. These calculations have been done using a master–slave programming paradigm on a coarse-grain, multiple instruction multiple data, shared-memory architecture. In this study, by avoiding the assembly of the global stiffness matrix, the memory requirement and computation time has been reduced. The results of the current study show that the parallel computing technique is a valuable tool for solving computationally intensive topology optimization problems.     

Integration of design and manufacturing for structural shape optimization

This paper presents an integrated design and manufacturing approach that supports shape optimization of structural components. The approach starts from a primitive concept stage, where boundary and loading conditions of the structural component are given to the designer. Topology optimization is conducted for an initial structural layout. The discretized structural layout is smoothed using parametric B-Spline surfaces. The B-Spline surfaces are imported into a CAD system to construct parametric solid models for shape optimization. Virtual manufacturing (VM) techniques are employed to ensure that the optimized shape can be manufactured at a reasonable cost. The solid freeform fabrication (SFF) system fabricates physical prototypes of the structure for design verification. Finally, a computer numerical control (CNC) machine is employed to fabricate functional parts as well as mold or die for mass production of the structural component. The main contribution of the paper is incorporating manufacturing into the design process, where manufacturing cost is considered for design. In addition, the overall design process starts from a primitive stage and ends with functional parts. A 3D tracked vehicle roadarm is employed throughout this paper to illustrate the overall design process and various techniques involved.     

Parametric design with neural network relationships and fuzzy relationships considering uncertainties

This research introduces a new parametric design approach with neural network relationships and fuzzy relationships considering uncertainties. In this work, parameters are associated by a hybrid parameter relationship network. In addition to deterministic parameters and relationships, non-deterministic parameters (e.g., random parameters and fuzzy parameters) and non-deterministic relationships (e.g., neural network relationships and fuzzy relationships) can also be modeled in this network. Changes of parameter values and their uncertainties are propagated through this network. Two types of optimization methods, reliability based design optimization and possibility based design optimization, are employed to identify the optimal design considering objective random uncertainties and subjective fuzzy uncertainties. A computer system has been implemented and used for the optimal design of a solid oxide fuel cell (SOFC) system considering uncertainties.     

Numerical Simulation of a Weakly Nonlinear Model for Water Waves with Viscosity

The potential flow equations which govern the free-surface motion of an ideal fluid (the water wave problem) are notoriously difficult to solve for a number of reasons. First, they are a classical free-boundary problem where the domain shape is one of the unknowns to be found. Additionally, they are strongly nonlinear (with derivatives appearing in the nonlinearity) without a natural dissipation mechanism so that spurious high-frequency modes are not damped. In this contribution we address the latter of these difficulties using a surface formulation (which addresses the former complication) supplemented with physically-motivated viscous effects recently derived by Dias et al. (Phys. Lett. A 372:1297–1302, 2008). The novelty of our approach is to derive a weakly nonlinear model from the surface formulation of Zakharov (J. Appl. Mech. Tech. Phys. 9:190–194, 1968) and Craig and Sulem (J. Comput. Phys. 108:73–83, 1993), complemented with the viscous effects mentioned above. Our new model is simple to implement while being both faithful to the physics of the problem and extremely stable numerically.     

基于隐式参数化耦合模型的车身集成优化

为提高车身截面优化效率,基于SFE CONCEPT构建白车身关注部位的隐式参数化模型,并与白车身有限元非参数化模型进行耦合,使参数化模型与有限元模型耦合边界处的连接关系随截面变化自动更新,通过试验验证耦合模型的有效性。基于试验设计（design of experiments, DOE）方法、近似模型、多目标优化等策略,对白车身耦合模型进行刚度和模态等多学科集成优化,实现车身局部结构快速轻量化设计。     

The integration of design of experiments, surrogate modeling and optimization for thermoscience research

This paper presents an integrated approach for the solution of complex optimization problems in thermoscience research. The cited approach is based on the design of computational experiments (DOE), surrogate modeling, and optimization. The DOE/surrogate modeling techniques under consideration include: A-optimal/classical linear regression, Latin hypercube/artificial neural networks, and Latin hypercube/Sugeno-type fuzzy models. These techniques are coupled with both local (modified Newtons method) and global (genetic algorithms) optimization methods. The proposed approach proved to be an effective, efficient and robust modeling and optimization tool in the context of a case study, and holds promise for use in larger scale optimization problems in thermoscience research.     

Design optimization through parallel-generated surrogate models, optimization methodologies and the utility of legacy simulation software

This paper presents the motivation development and an application of a unique methodology to solve industrial optimization problems, using existing legacy simulation software programs. The methodology is based on approximation models generated with the utility of design of experiments methodologies and response surface methods applied on high-fidelity simulations, coupled together with classical optimization methodologies. Several DOE plans are included, in order to be able to adopt the appropriate level of detail. The approximations are based on stochastic interpolation techniques, or on classical least squares methods. The optimization methods include both local and global techniques. Finally, an application from the plastic molding industry (process simulation) demonstrates the methodology and the software package. Received December 30, 2000     

Parametric shape optimization techniques based on Meshless methods: A review

In the product development process, structural optimization plays vital role because it deals with size, shape and topology of the structures. However, structural performance greatly depends on its geometric shape and hence structural shape optimization has remained one of the most active research areas since early 1970s. Conventional parametric shape optimization technique employs grid-based numerical tools like FEM and BEM for structural analysis, which experiences some innate limitations like mesh distortion and frequent remeshing, element locking and poor approximation while dealing with large shape changes during the optimization process. Meshless Methods (MMs) can alleviate these issues when used as a structural analysis tool in shape optimization. In last two decades, MMs have been explored for structural shape optimization along with various deterministic and stochastic optimization algorithms. The objective of present work is twofold, first is to review advanced parametric shape optimization techniques which are based on MMs like Element Free Galerkin (EFG) method and Reproducing Kernel Particle Method (RKPM) for linear elastic, thermoelastic, hyperelastic, frictional contact and structure dynamics optimization problems and second is to emphasize benefits of meshless techniques in shape optimization. Based on the review, the article presents some critical observations including Design Sensitivity Analysis (DSA) in meshless environment, numerical integration techniques in MMs and benefits of coupled FEM-MM approach in shape optimization. At the end, promising future research directions in shape optimization field based on MMs are presented along with concluding remarks.     

An evaluation of back-propagation neural networks for the optimal design of structural systems: Part I. Training procedures

Design optimization using approximations based on feed-forward back-propagation neural network is the topic of much recent research. The neural network schemes that have been proposed in the literature for optimal design of structural systems differ in their architecture and training procedures. Furthermore, their utility vis-a-vis classical optimization techniques is not always clear. A systematic comparison of the efficiency and accuracy of the neural network-based solution schemes to classical structural optimization techniques is the aim of this and the companion paper. In this paper, the neural network training procedures used in the present evaluation are described in detail. When using first-order nonlinear programming algorithms with neural networks, the ability to approximate derivatives is important. Therefore, mainly for completeness of evaluation, two new training methods that use the derivative information are proposed in addition to the now common function-based training method. The first method uses the derivatives to create additional training points in the vicinity of the original points, based on Taylor's series expansion. The second method attempts to minimize the error in derivatives while imposing the error in output functions as constraint. Expressions for analytical derivatives are derived for both function-based and derivative-based training. Significant savings in computational time are reported when calculating derivatives using built-in analytical derivatives instead of using finite difference derivatives. In the companion paper the proposed methods are applied to solve five optimization problems with varying degree of complexity. Approximately 1100 test cases are executed in the companion paper to compare the accuracy and efficiency of neural network-based optimization with the classical approaches.     

BOSS QUATTRO: an open system for parametric design

During the two past decades, engineers have shown growing interest in automatic structural optimization techniques. These were first used to solve analytical problems and were then rapidly adapted to structural sizing problems coupled to finite element analysis software. Moreover, shape optimization problems featuring complex CAD systems were addressed in the early 90’s. This article describes the capabilities of the optimization product developed by SAMTECH S.A. in Liège, Belgium. One will read here what led the company and a group of research engineers at LTAS (Laboratoire des Techniques Aéronautiques et Spatiales, Université de Liège) to the achievement of a so-called open system for parametric design. The BOSS/Quattro package is a general-purpose design program. It includes several “engines”: optimization, parametric studies, “Monte-Carlo” studies, “design of experiments” and updating. It is interesting to note that the “engines” can be easily mixed through the graphical user interface (GUI), and, this, with several analysis models. Received December 30, 2000     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

基于隐式参数化的车身多学科轻量化设计

为实现整车综合性能的快速方案验证和优化设计,在新车型设计阶段构建车身隐式参数化模型,并对其进行模态、刚度和安全等综合性能计算,验证参数化模型的有效性。基于灵敏度分析、试验设计（design of experiments, DOE）方法和近似模型优化等策略,对某白车身进行多学科轻量化设计。优化设计结果表明,白车身的模态、刚度和安全性能均满足设计要求。     

Metamodels for Computer-based Engineering Design: Survey and recommendations

The use of statistical techniques to build approximations of expensive computer analysis codes pervades much of today’s engineering design. These statistical approximations, or metamodels, are used to replace the actual expensive computer analyses, facilitating multidisciplinary, multiobjective optimization and concept exploration. In this paper, we review several of these techniques, including design of experiments, response surface methodology, Taguchi methods, neural networks, inductive learning and kriging. We survey their existing application in engineering design, and then address the dangers of applying traditional statistical techniques to approximate deterministic computer analysis codes. We conclude with recommendations for the appropriate use of statistical approximation techniques in given situations, and how common pitfalls can be avoided.     

Surface hardening of commercially pure titanium by laser nitriding: Response surface analysis

In this work laser surface nitriding was performed to enhance wear and erosion resistance of pure titanium by increasing its surface hardness while keeping the strength and ductility of the core for static and dynamic loading resistance. The Response Surface Methodology (RSM) in the Design of Experiment (DOE) statistical method is adopted in this research work to perform the experimental design, analysis and optimization of the laser nitriding process. Continuous wave CO₂ laser was used to melt the surface, and then nitrogen gas was incorporated into the melted pool to produce hard TiN. The Response Surface Design was first created using MINTAB program, and then the laser nitriding process was performed according to the planned design. Microhardness is then measured for each sample, which represents the response, and incorporated into the design matrix. Results are then analyzed and a RSM model was developed and verified. The model is then used to perform parametric study and optimization. The maximum measured microhardness based on the original RSM design was 1382 HV_0.15. However, based on the model prediction, the optimal process parameters settings were found to be as: 2.84 kW laser power, 5 mm/s scanning speed and 2076 l/h nitrogen flow rate which would result in a maximum microhardness of approximately 1920 HV_0.15.     

A Comparative study on multiobjective reliable and robust optimization for crashworthiness design of vehicle structure

Design optimization without considering uncertainties of system variables and parameters can be problematic in real life. In order to take into account the effect of uncertainties, reliable and robust design schemes have proven effective, but limited studies have been reported to compare their difference in a multiobjective framework. This paper takes a typical vehicle structure subject to offset frontal crashing scenario as an example to compare reliable and robust designs with their deterministic counterpart. The thicknesses of some key components of vehicle frontal structures were selected as design variables, the vehicle weight and energy absorption as the objectives, deceleration and firewall intrusion as the constraints. The deterministic multiobjective optimization problem was first solved by adopting Design of Experimental (DOE), metamodels and Non-dominated Sorting Genetic Algorithm II (NSGA-II). Take into account the uncertainties, a Monte Carlo Simulation (MCS) is adopted to generate random distributions of the objective and constraint functions for each design. For the reliability-based optimization the desired reliabilities of 90 %, 95 % and 99 % are considered, respectively. For the robustness-based optimization, two different formulation strategies are adopted. The optimization showed that the reliable and robust Pareto fronts are shifted away from their deterministic counterpart due to uncertainties. The different Pareto fronts yielded from the deterministic, reliable and robust designs are compared to provide some quantitative insights into how to apply these different design schemes for resolving uncertainty problems. It is shown that, compared with the baseline design, the optimizations enhance the crashworthiness of vehicle, though more conservative solutions could have been generated from the reliable and robust optimizations.     

Parameter selection in synchronous and asynchronous deterministic particle swarm optimization for ship hydrodynamics problems

Deterministic optimization algorithms are very attractive when the objective function is computationally expensive and therefore the statistical analysis of the optimization outcomes becomes too expensive. Among deterministic methods, deterministic particle swarm optimization (DPSO) has several attractive characteristics such as the simplicity of the heuristics, the ease of implementation, and its often fairly remarkable effectiveness. The performances of DPSO depend on four main setting parameters: the number of swarm particles, their initialization, the set of coefficients defining the swarm behavior, and (for box-constrained optimization) the method to handle the box constraints. Here, a parametric study of DPSO is presented, with application to simulation-based design in ship hydrodynamics. The objective is the identification of the most promising setup for both synchronous and asynchronous implementations of DPSO. The analysis is performed under the assumption of limited computational resources and large computational burden of the objective function evaluation. The analysis is conducted using 100 analytical test functions (with dimensionality from two to fifty) and three performance criteria, varying the swarm size, initialization, coefficients, and the method for the box constraints, resulting in more than 40,000 optimizations. The most promising setup is applied to the hull-form optimization of a high speed catamaran, for resistance reduction in calm water and at fixed speed, using a potential-flow solver.     

An open platform of shape design optimization for shell structure

A general platform built on a computer-aided design (CAD) system is developed for parameterized shape design optimization of shell structure. Within the platform, parameterized surface modeling and computer-aided engineering (CAE) applications are embedded and seamlessly integrated with the CAD system through its application programming interface (API). Firstly, instead of the CAD system inherent surface modeling, a parameterized surface modeling for shell structure is fulfilled through integrating with parametric solid modeling of the CAD system. Thus, any dimensions for parametric solid modeling can be used to control shape modification of shell structure and serve as design variables for shape design optimization. Secondly, seamless integration of geometry modeling and finite-element modeling for shell structure is implemented. Finally, with integrated procedures of finite-element analysis and optimization algorithms, a general platform for parameterized shape optimization of shell structure is realized. Numerical examples are presented, and the results validate the effectiveness and efficiency of the platform. A shorten version of this paper was presented to the 7th World Congress of Computation Mechanics (WCCM 2006), July 16–22, 2006, Los Angeles, CA, USA.     

On an alternative approach to stress constraints relaxation in topology optimization

The paper deals with the imposition of local stress constraints in topology optimization. The aim of the work is to analyze the performances of an alternative methodology to the ε-relaxation introduced in Cheng and Guo (Struct Optim 13:258–266, 1997), which handles the well-known stress singularity problem. The proposed methodology consists in introducing, in the SIMP law used to apply stress constraints, suitable penalty exponents that are different from those that interpolate stiffness parameters. The approach is similar to the classical one because its main effect is to produce a relaxation of the stress constraints, but it is different in terms of convergence features. The technique is compared with the classical one in the context of stress-constrained minimum-weight topology optimization. Firstly, the problem is studied in a modified truss design framework, where the arising of the singularity phenomenon can be easily shown analytically. Afterwards, the analysis is extended to its natural context of topology bidimensional problems.