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Based on course-tested material, this rigorous yet accessible graduate textbook covers both fundamental and advanced optimization theory and algorithms. It covers a wide range of numerical methods and topics, including both gradient-based and gradient-free algorithms, multidisciplinary design optimization, and uncertainty, with instruction on how to determine which algorithm should be used for a given application. It also provides an overview of models and how to prepare them for use with numerical optimization, including derivative computation. Over 400 high-quality visualizations and numerous examples facilitate understanding of the theory, and practical tips address common issues encountered in practical engineering design optimization and how to address them. Numerous end-of-chapter homework problems, progressing in difficulty, help put knowledge into practice. Accompanied online by a solutions manual for instructors and source code for problems, this is ideal for a one- or two-semester graduate course on optimization in aerospace, civil, mechanical, electrical, and chemical engineering departments.

Based on course-tested material, this rigorous yet accessible graduate textbook covers both fundamental and advanced optimization theory and algorithms. It covers a wide range of numerical methods and topics, including both gradient-based and gradient-free algorithms, multidisciplinary design optimization, and uncertainty, with instruction on how to determine which algorithm should be used for a given application. It also provides an overview of models and how to prepare them for use with numerical optimization, including derivative computation. Over 400 high-quality visualizations and numerous examples facilitate understanding of the theory, and practical tips address common issues encountered in practical engineering design optimization and how to address them. Numerous end-of-chapter homework problems, progressing in difficulty, help put knowledge into practice. Accompanied online by a solutions manual for instructors and source code for problems, this is ideal for a one- or two-semester graduate course on optimization in aerospace, civil, mechanical, electrical, and chemical engineering departments.

The optimal design of multidisciplinary systems under uncertainty involves three types of iterative analyses- multidisciplinary analyses, reliability analyses, and optimization. If these processes are carried out in a nested manner in reliability based multidisciplinary design optimization (RBMDO), the computational expense can be enormous. Computationally efficient simultaneous analysis and design (SAND) or All-AtOnce methods have been developed earlier for deterministic MDO. Single loop RBDO methods have also been developed to integrate reliability analysis and optimization in a single loop. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints. The formulation allows for solution with a single design optimization loop by using the KKT optimality conditions of the first-order reliability method (FORM) as constraints in the design optimization along with disciplinary consistency constraints. Alternative solution processes which remove the FORM MPP’s from the decision space and many constraints from the problem are developed. The alternative solution methods are more efficient and more robust than using nonlinear programming to solve the single loop formulation. A numerical example is solved with the two proposed solution algorithms. I. Introduction In the design of multidisciplinary systems it is often necessary to make decisions under uncertainty. This requires that multidisciplinary systems modeling be integrated with uncertainty analysis and optimization. A common example of such an engineering design application would be the design of an aircraft wing under uncertain aerodynamic loading conditions, with multiple structural components with uncertain capacities, such that the entire structural system achieves a prescribed level of reliability. The problem has coupled physics, in that the pressure distribution is dependent upon the deflection, and the deflection is dependent on the pressure distribution. Such designs have been performed using multidisciplinary optimization (MDO). Other examples where MDO may be applied include electronics packaging, in which the disciplines of heat transfer and circuit analysis are coupled, the design of new types of aircraft and integrating them into an existing fleet of aircraft, and even the management of interconnected homeland infrastructures. While MDO has matured with the development of different types of algorithms for various types of problems and design organizations, there has been little work to incorporate reliability analysis into MDO. However, system and component reliabilities are often a major concern in the design of multidisciplinary systems. MDO can be computationally expensive due to the need to achieve consistency across multiple disciplinary analysis codes; each disciplinary analysis code may be computationally expensive to run. Further, the optimal design of multidisciplinary systems under uncertainty involves three types of iterative procedures- multidisciplinary analyses, reliability analyses, and optimization. The integration of these three procedures is referred to in this paper as reliability based multidisciplinary design optimization (RBMDO). If these processes are carried out simultaneously, the computational expense can be enormous. Computational savings can often be achieved with simultaneous analysis and design (SAND) or All-At-Once methods, which allow for the solution of the multidisciplinary analyses and the optimization in one single loop. Single loop RBDO methods allow for the integration of reliability analysis and optimization in a single loop. Applying SAND and single-loop RBDO together may allow for large computational savings when applied to problems in RBMDO. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints based on the SAND method for MDO and by use of the KKT conditions of the first order reliability method (FORM). The resulting formulation may be solved as a standard nonlinear programming problem. While the single loop formulation presented in this paper is a solution of the RBMDO problem, robustness and efficiency are improved through two improved algorithms which can remove the MPP’s from the decision space, thereby significantly reducing the optimizer’s workload.

Model-Based Systems Engineering (MBSE) supports the system-level design of complex products effectively. Currently, system design and optimization for complex products are two distinct processes that must be executed using different software or platforms, involving intricate data conversion processes. Applying multidisciplinary optimization to validate system optimization often necessitates remodeling the optimization objects, which is time-consuming, labor-intensive, and highly error-prone. A critical activity in systems engineering is identifying the optimal design solution for the entire system. Multidisciplinary Design Optimization (MDO) and reliability analysis are essential methods for achieving this. This paper proposes a SysML-based multidisciplinary reliability design optimization modeling method. First, by analyzing the definitions and mathematical models of multidisciplinary reliability design optimization, the SysML extension mechanism is employed to represent the optimization model based on SysML. Next, model transformation techniques are used to convert the SysML optimization model generated in the first stage into an XML description model readable by optimization solvers. Finally, the proposed method’s effectiveness is validated through an engineering case study of an in-vehicle environmental control integration system. The results demonstrate that this method fully utilizes SysML to express MDO problems, enhancing the efficiency of design optimization for complex systems. Engineers and system designers working on complex, multidisciplinary projects can particularly benefit from these advancements, as they simplify the integration of design and optimization processes, facilitating more reliable and efficient product development.

A blowout preventer (BOP) is a large valve that encases an oil well on the surface. The valve may be closed while drilling if overpressure from a reservoir causes formation fluids such as oil and natural gas to back up within the wellbore and endanger the rig. A pipe ram is one of the critical components in the BOP system, which is designed to seal around a drill pipe, restricting the flow in the annulus between the outside of the drill pipe and the wellbore. The pipe ram design family, which contains hundreds of pipe ram designs, intends to cover different configurations to suit a variety of dill pipe sizes. To assure their structural integrity under service loads, these pipe rams need to be designed to meet the stress requirements per the American Petroleum Institute (API) Specification 16A and ASME (American Society of Mechanical Engineers) Boiler and Pressure Vessel Code Section VIII, Division 2. With a conventional manual design workflow, it would take a significant amount of time and effort to complete all the designs for the product family with code compliance. Therefore, a scalable solution is highly desirable for delivery of customer-needed ram blocks with much shorter lead time. MDO (Multi-Disciplinary Optimization) involves multi-code integration, CAE (Computer-Aided Engineering) workflow automation, design space exploration, and optimization. MDO-enabled, automated design optimization is becoming increasingly popular in both scientific and engineering communities. In this paper, a methodology for integration of CAD (Computer Aided Design), FEA (Finite Element Analysis), cost, and optimization packages to enable FEA automation and design optimization is presented. A BOP pipe ram is adopted as the use case. The ram block geometry was parameterized before being imported into an FEA package. The FEA workflow was automated such that once a set of geometric parameters are given, the preprocessing, solving, and postprocessing steps can be automatically completed. As part of the FEA postprocessing, stress linearization analysis per the API and ASME BPVC codes has also been automated, which had never been done in the past. A manufacturing cost analysis package can also be used to consume the parameterized geometry for automatic manufacturability assessment and cost predictions. The stress analysis and cost analysis workflows are conducted separately but also orchestrated by an MDO package. Reports that contain the analysis results are sequentially generated for various design permutations. The MDO-enabled automated design and cost analysis approach could substantially enhance efficiency and consistency in performing FEA and cost studies and producing analysis reports. It is also the backbone for automated design optimization, which could significantly improve the product performance and reliability and, meanwhile, minimize the development and product cost.

In order to design a vehicle body with high strength and high stiffness, a multi-disciplinary design process should include careful consideration of multi-disciplinary design constraints to properly account for vehicle static stiffness (bending/torsional), durability, Noise/Vibration/Harshness (NVH), crash worthiness, light weight vehicle structure during the early stage of vehicle design process. With this approach, fast development of new vehicle body structures can be achieved with minimal number of iterations to match the conflicting design goals from each discipline. In the current research, a multi-disciplinary design optimization (MDO) based on a meta model is developed and refined to apply for the design of body structure. In an effort to apply the MDO for vehicle body structure, 4 phase procedures were established in the current research. In Phase I, a base model is created. In Phase II, an effect analysis is carried out. In Phase III, a meta model is created. Finally in Phase IV, using the optimization algorithm, the meta model created in Phase III is eventually refined through the process of optimization. In this research, static stiffness (bending / torsional), dynamic stiffness (1st torsion mode) were used for constrained conditions and the mass minimization was the object function for optimization.

Battery thermal management system (BTMS) is a complex and highly integrated system, which is used to control the battery thermal conditions in electric vehicles (EVs). The BTMS consists of many subsystems that belong to different disciplines, which poses challenges to BTMS optimization using conventional methods. This paper develops a general variable fidelity-based multidisciplinary design optimization (MDO) architecture and optimizes the BTMS by considering different systems/disciplines from the systemic perspective. Four subsystems and/or subdisciplines are modeled, including the battery thermodynamics, fluid dynamics, structure, and lifetime model. To perform the variable fidelity-based MDO of the BTMS, two computational fluid dynamics (CFD) models with different levels of fidelity are developed. A low fidelity surrogate model and a tuned low fidelity model are also developed using an automatic surrogate model selection method, the concurrent surrogate model selection (COSMOS). An adaptive model switching (AMS) method is utilized to realize the adaptive switch between variable-fidelity models. The objectives are to maximize the battery lifetime and to minimize the battery volume, the fan's power, and the temperature difference among different cells. The results show that the variable-fidelity MDO can balance the characteristics of the low fidelity mathematical models and the computationally expensive simulations, and find the optimal solutions efficiently and accurately.

NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session 1432 A Combined Electrical and Mechanical Engineering Senior Design Laboratory M. E. Parten and D. L. Vines T. T. Maxwell and J. C. Jones Electrical Engineering Mechanical Engineering Texas Tech University Lubbock, TX 79409 Abstract This paper describes a multidisciplinary capstone design laboratory course offered in the Electrical and Mechanical Engineering Departments at Texas Tech University. The course uses projects from industry, research efforts and other faculty initiatives. The projects for the course come from industry, research efforts and other faculty initiatives. The courses are team taught by both departments. This paper describes the structure of the program along with example projects and results. Both positive and negative aspects of the program are presented. Introduction In the fall semester of 1994 the Electrical and Mechanical Engineering Departments at Texas Tech University began a multidisciplinary senior design project laboratory program. Two courses were established by integrating the Electrical Engineering Department's Senior Project Laboratory courses (two 3-semester credit hour courses) with the Mechanical Engineering Department's Design I and II capstone design courses (two 3-semester credit hour courses). The Electrical Engineering Department has a long history of project laboratories.1-5 The Mechanical Engineering Department has been involved in alternative fueled vehicles for a number of years. Both departments had worked together on a number of special projects and felt the need, as have many others6-11, for an increased interdisciplinary program for engineering students. The goals of these new courses were: to have the students develop an understanding of engineering design projects from recognition of a need and definition of design objectives through completion of the project to foster student creativity to broaden the students concept of engineering problems to include other engineering disciplines and other nonengineering factors that have an impact on the final problem solution to provide a unique educational experience for students on project teams to enhance the students communication skills The projects for the course come from industry, research efforts and other faculty initiatives.

Research on multidisciplinary fatigue optimization design method in structural design of high speed train

The design of multidiscipli nary systems (such as aircraft, automobiles, and others) often requires an iterative cycle that includes a design initialization, a system analysis, a sensitivity analysis, and design optimization. This design cycle is standard in the field of Multidisciplinary Optimization (MDO) and has often been referred to in literature as the Multiple Discipline-Feasible (MDF) approach. The name stems from the fact that complete multidisciplinary feasibility is maintained each and every design cycle. The drawback of MDF is that it can be a timely and a costly procedure. Numerous researchers have developed alternate means for posing and subsequently solving the multidisciplinary design problem. One such solution procedure has been referred to both as Simultaneous Analysis and Design (SAND) and All-at-Once (AAO), and treats the entire multidisciplinary design cycle as one large optimization problem. Another alternate solution procedure lias been referred to as Individual-Discipline-Feasible (DDF); this procedure exhibits characteristics which lie in between the two extremes exemplified by MDF and AAO. IDF assures that each individual discipline is feasible on every design cycle, while driving the entire system (all disciplines) towards multidisciplinary feasibility. The present work will present a rigorous numerical comparison of these solution strategies over a wide variety of problem sizes and complexities. The multidisciplinary design test problems that are used for these comparisons are generated by a robust simulation tool called CASCADE. * Research Assistant, Student member, AIAA | Associate Professor, Member AIAA Background and Motivation Concurrent engineering is a systematic approach to the integrated, simultaneous design of products and their related processes. Many of the recently developed capabilities to address concurrent engineering have stemmed from the emerging area of Multidiscipli nary Optimization, or MDO. The origins of MDO can be traced back to the early 1980's, where Sobieski14 used a linear decomposition approach to subdivide the design of a large engineering system into a grouping of related and more manageable subsystems. However, such a decomposition often results in a grouping of subsystems which cannot be placed into a definitive top-down hierarchy. The resultant decomposition grouping is typically hybrid-hierarchic in nature as shown in the example decomposition of Figure 1.

<p>Multidisciplinary design optimization (MDO) refers to the process of designing systems characterized by the interaction of multiple interconnected disciplines. High-fidelity MDO usually requires large computational resources due to the computational cost of achieving multidisciplinary consistent solutions by coupling high-fidelity physics-based solvers. Gradient-based minimization algorithms are generally applied to find local minima, due to their efficiency in solving problems with a large number of design variables. This represents a limitation to performing global MDO and integrating black-box type analysis tools, usually not providing gradient information. The latter issues generally inhibit a wide use of MDO in complex industrial applications.</p><p>An architecture named multi-criterion adaptive sampling MDO (MCAS-MDO) is presented in the current research for complex simulation-based applications. This research aims at building a global derivative-free optimization tool able to employ high-fidelity/expensive black-box solvers for the analysis of the disciplines. MCAS-MDO is a surrogate-based architecture featuring a variable level of coupling among the disciplines and is driven by a multi-criterion adaptive sampling (MCAS) assessing coupling and sampling uncertainties. MCAS uses the dynamic radial basis function surrogate model to identify the optimal solution and explore the design space through parallel infill of new solutions.</p><p>The MCAS-MDO is tested versus a global derivative-free multidisciplinary feasible (MDF) approach, which solves fully-coupled multidisciplinary analyses, for two analytical test problems. Evaluation metrics include number of function evaluations required to achieve the optimal solution and sample distribution. The MCAS-MDO outperforms the MDF showing a faster convergence by clustering refined function evaluations in the optimum region.</p><p>The architecture is applied to a steady fluid-structure interaction (FSI) problem, namely the design of a tapered three-dimensional carbon fiber-reinforced plastic hydrofoil for minimum drag. The objective is the design of shape and composite material layout subject to hydrodynamic, structural, and geometrical constraints. Experimental data are available for the original configuration of the hydrofoil and allow validating the FSI analysis, which is performed coupling computational fluid dynamics, solving the Reynolds averaged Navier-Stokes equations, and finite elements, solving the structural equation of elastic motion. Hydrofoil forces, tip displacement, and tip twist are evaluated for several materials providing qualitative agreement with the experiments and confirming the need for the two-way versus one-way coupling approach in case of significantly compliant structures.</p><p>The free-form deformation method is applied to generate shape modifications of the hydrofoil geometry. To reduce the global computational expense of the optimization, a design space assessment and dimensionality reduction based on the Karhunen–Loève expansion (KLE) is performed off-line, i.e. without the need for high-fidelity simulations. It provides with a selection of design variables for the problem at hand through basis rotation and re-parametrization. By using the KLE, an efficient design space is identified for the current problem and the number of design variables is reduced by 92%.</p><p>A sensitivity analysis is performed prior to the optimization to assess the variability associated with the shape design variables and the composite material design variable, i.e. the fiber orientation. These simulations are used to initialize the surrogate model for the optimization, which is carried out for two models: one in aluminum and one in composite material. The optimized designs are assessed by comparison with the original models through evaluation of the flow field, pressure distribution on the body, and deformation under the hydrodynamic load. The drag of the aluminum and composite material hydrofoils is reduced by 4 and 11%, respectively, increasing the hydrodynamic efficiency by 4 and 7%. The optimized designs are obtained by evaluating approximately 100 designs. The quality of the results indicates that global derivative-free MDO of complex engineering applications using expensive black-box solvers can be achieved at a feasible computational cost by minimizing the design space dimensionality and performing an intelligent sampling to train the surrogate-based optimization.</p>

Design engineers and decision-makers across various fields are constantly working to make optimal design decisions for multidisciplinary engineering systems in an effort to improve performance and reduce costs. The multiple disciplines that decision-makers are forced to consider can range from different physical components of a system, to competing physical phenomena influencing a component (e.g. flow forces and structural strength), to completely separate models of interest to a system (e.g. engineering performance and lifecycle cost). The common element that all these decision-making scenarios share is the presence of couplings between the considered disciplines (or subsystems). How the values for these coupling parameters are determined within a decision-making or optimization framework is the subject of countless research efforts. At present the multidisciplinary design optimization (MDO) community has settled on a few proven techniques such Collaborative Optimization (CO) and Analytic Target Cascading (ATC). However, current MDO techniques have issues that limit their effectiveness in solving various MDO problems. Many of these strategies require close coordination between subsystem optimization solvers, require significant effort by decision-makers to pose their problems in a suitable format, and/or can have large computational efficiency problems due to the fact that they involve solving nested optimization problems. In an effort to alleviate some of these issues and make MDO easier to implement and more computationally efficient, a new sequential MDO algorithm called Cooperative Design Optimization (CDO) is proposed. The CDO approach functions through a series of subsystem optimizations using a successively smaller cooperation space. The cooperation space is analogous to the design space of a traditional optimization problem, but includes only the coupling parameters that are a factor in multiple subsystems. A single iteration of the approach can be thought of as a negotiation between all of the subsystems regarding the boundaries of the cooperation space. To facilitate this cooperation, each subsystem optimization problem is restructured as a multi-objective optimization problem so that a Pareto set of optimal solutions, or agreeable design alternatives, are produced. As a result, after all subsystem optimizations are accomplished, each subsystem’s obtained Pareto optimal solution sets are compared with respect to the coupling parameters and new bounds in the cooperation space are determined for the next iteration. This process allows each subsystem to be optimized completely independently within the boundaries of the agreed upon cooperation space while coordination is achieved through an analysis of obtained optimal solutions for each subsystem. Iterations, or negotiations, are repeated until an acceptable solution or set of solutions is obtained for all included subsystems through this cooperative process.

A modern approach to graduate flight dynamics, stability, and control courses

Multidisciplinary design optimization (MDO) is concerned with solving design problems involving coupled numerical models of complex engineering systems. While various MDO software frameworks exist, none of them take full advantage of state-of-the-art algorithms to solve coupled models efficiently. Furthermore, there is a need to facilitate the computation of the derivatives of these coupled models for use with gradient-based optimization algorithms to enable design with respect to large numbers of variables. In this paper, we present the theory and architecture of OpenMDAO, an open-source MDO framework that uses Newton-type algorithms to solve coupled systems and exploits problem structure through new hierarchical strategies to achieve high computational efficiency. OpenMDAO also provides a framework for computing coupled derivatives efficiently and in a way that exploits problem sparsity. We demonstrate the framework’s efficiency by benchmarking scalable test problems. We also summarize a number of OpenMDAO applications previously reported in the literature, which include trajectory optimization, wing design, and structural topology optimization, demonstrating that the framework is effective in both coupling existing models and developing new multidisciplinary models from the ground up. Given the potential of the OpenMDAO framework, we expect the number of users and developers to continue growing, enabling even more diverse applications in engineering analysis and design.

The implementation of a general and flexible method of formulating problems of mathematical programming in structural and multidisciplinary optimization systems is described. The method, which is based on the so-called bound formulation, enables the formulation and solution of problems involving local, integral, mintmax, maxlmin and possibly non-differentiable user-defined functions in any conceivable mix and using user-defined mathematical functions. The computer implementation involves a parser capable of interpreting and performing symbolic dfierentiation of the user defined functions. A sample problem of thermo-elasticity is given. It is concluded that this approach is well suited for multidisciplinary design optimization systems. 1. Motivation One of the great challenges of research in design automation is the development of practically applicable systems for multidisciplinary design optimization. Such systems will, if they are truly generally applicable, cause a revolution in the way design laboratories work. Several papers with the focus on system design rather than single applications have emerged in recent years. Some examples are Esping (1986), Rasmussen (1989), Papalarnbros & Chirehdast (1990), Choi & Chang (1991), Botlun et. al. (1 992), Rasmussen et. al. (1993), and Cardoso & Santos (1 993). Obtaining the necessary generality, however, is a very difficult task. Designing even simple components involves many different considerations. For the sake of the argument, we may categorize them into groups of increasing complexity: The functional behavior, for instance weight, energy consumption, power output, efficiency, eigenfrequencies, strength, stiffness, pressures, velocities, accelerations, temperatures, electric and magnetic properties, etc. These qualities can often be modeled mathematically and analyzed by means of digital computers. The fabricational behavior, for instance the cost of machining, the necessary production facilities, the necessary manpower, stock and inventory, financial requirements, etc. These properties are usually very difficult to analyze because of the many degrees of freedom in choice of production planning and processes, the dependency of other products manufactured simultaneously, and even the dependence on the economic boundary conditions formed by the outside world. The aesthetic properties that are impossible to evaluate objectively, but are none the less tightly connected with the functional and fabricational properties. For instance, the shape of the hull of a ship determines strength, stiffness and drag properties, but it also forms an important part of the visual impression. In the development of multidisciplinary systems, we have only begun to cover the first category of product properties, so it may be argued that there is still a long way to go. At the Institute of Mechanical Engineering, Aalborg University, Denmark, a system for multidisciplinary design optimization is under development. The system is called 'Associate Professor of Mechanical Engineering '+~esearch Assistant Copyright O 1994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.