scholarly journals Approximating solution spaces as a product of polygons

2021 ◽  
Author(s):  
Helmut Harbrecht ◽  
Dennis Tröndle ◽  
Markus Zimmermann
Keyword(s):  
Design Space ◽  
Complete Solution ◽  
Cartesian Product ◽  
Solution Space ◽  
Monte Carlo Sampling ◽  
High Dimensional ◽  
Two Dimensional ◽  
Design Work ◽  
Design Variables ◽  
Linear Problems

AbstractSolution spaces are regions of good designs in a potentially high-dimensional design space. Good designs satisfy by definition all requirements that are imposed on them as mathematical constraints. In previous work, the complete solution space was approximated by a hyper-rectangle, i.e., the Cartesian product of permissible intervals for design variables. These intervals serve as independent target regions for distributed and separated design work. For a better approximation, i.e., a larger resulting solution space, this article proposes to compute the Cartesian product of two-dimensional regions, so-called 2d-spaces, that are enclosed by polygons. 2d-spaces serve as target regions for pairs of variables and are independent of other 2d-spaces. A numerical algorithm for non-linear problems is presented that is based on iterative Monte Carlo sampling.

On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems

2017 ◽  
Vol 4 (2) ◽  
Author(s):  
Stefan Erschen ◽  
Fabian Duddeck ◽  
Matthias Gerdts ◽  
Markus Zimmermann
Keyword(s):  
Complete Solution ◽  
Cartesian Product ◽  
Solution Space ◽  
Linear Functions ◽  
High Dimensional ◽  
Development Work ◽  
Interior Point Algorithm ◽  
Dimensional Solution ◽  
Design Work ◽  
Design Variables

In the early development phase of complex technical systems, uncertainties caused by unknown design restrictions must be considered. In order to avoid premature design decisions, sets of good designs, i.e., designs which satisfy all design goals, are sought rather than one optimal design that may later turn out to be infeasible. A set of good designs is called a solution space and serves as target region for design variables, including those that quantify properties of components or subsystems. Often, the solution space is approximated, e.g., to enable independent development work. Algorithms that approximate the solution space as high-dimensional boxes are available, in which edges represent permissible intervals for single design variables. The box size is maximized to provide large target regions and facilitate design work. As a result of geometrical mismatch, however, boxes typically capture only a small portion of the complete solution space. To reduce this loss of solution space while still enabling independent development work, this paper presents a new approach that optimizes a set of permissible two-dimensional (2D) regions for pairs of design variables, so-called 2D-spaces. Each 2D-space is confined by polygons. The Cartesian product of all 2D-spaces forms a solution space for all design variables. An optimization problem is formulated that maximizes the size of the solution space, and is solved using an interior-point algorithm. The approach is applicable to arbitrary systems with performance measures that can be expressed or approximated as linear functions of their design variables. Its effectiveness is demonstrated in a chassis design problem.

Identifying Key Parameters for Design Improvement in High-Dimensional Systems With Uncertainty

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Johannes Fender ◽  
L. Graff ◽  
H. Harbrecht ◽  
Markus Zimmermann
Keyword(s):  
Numerical Optimization ◽  
Solution Space ◽  
Design Criterion ◽  
Design Parameters ◽  
High Dimensional ◽  
Design Work ◽  
Good Design ◽  
Design Improvement ◽  
Front Structure ◽  
Key Parameter

Key parameters may be used to turn a bad design into a good design with comparatively little effort. The proposed method identifies key parameters in high-dimensional nonlinear systems that are subject to uncertainty. A numerical optimization algorithm seeks a solution space on which all designs are good, that is, they satisfy a specified design criterion. The solution space is box-shaped and provides target intervals for each parameter. A bad design may be turned into a good design by moving its key parameters into their target intervals. The solution space is computed so as to minimize the effort for design work: its shape is controlled by particular constraints such that it can be reached by changing only a small number of key parameters. Wide target intervals provide tolerance against uncertainty, which is naturally present in a design process, when design parameters are unknown or cannot be controlled exactly. In a simple two-dimensional example problem, the accuracy of the algorithm is demonstrated. In a high-dimensional vehicle crash design problem, an underperforming vehicle front structure is improved by identifying and appropriately changing a relevant key parameter.

Maximizing Flexibility for Complex Systems Design to Compensate Lack-of-Knowledge Uncertainty

2019 ◽  
Vol 5 (4) ◽  
Author(s):  
Marco Daub ◽  
Fabian Duddeck
Keyword(s):  
Complex Systems ◽  
Complete Solution ◽  
Solution Space ◽  
Systems Design ◽  
Rigid Wall ◽  
Total System ◽  
Independent Manner ◽  
Design Variables ◽  
Knowledge Uncertainty ◽  
Lack Of Knowledge

Abstract The consideration of uncertainty is especially important for the design of complex systems. Because of high complexity, the total system is normally divided into subsystems, which are treated in a hierarchical and ideally independent manner. In recent publications, e.g., (Zimmermann, M., and von Hoessle, J. E., 2013, “Computing Solution Spaces for Robust Design,” Int. J. Numer. Methods Eng., 94(3), pp. 290–307; Fender, J., Duddeck, F., and Zimmermann, M., 2017, “Direct Computation of Solution Spaces,” Struct. Multidiscip. Optim., 55(5), pp. 1787–1796), a decoupling strategy is realized via first the identification of the complete solution space (solutions not violating any design constraints) and second via derivation of a subset, a so-called box-shaped solution space, which allows for decoupling and therefore independent development of subsystems. By analyzing types of uncertainties occurring in early design stages, it becomes clear that especially lack-of-knowledge uncertainty dominates. Often, there is missing knowledge about overall manufacturing tolerances like limitations in production or subsystems are not even completely defined. Furthermore, flexibility is required to handle new requirements and shifting preferences concerning single subsystems arising later in the development. Hence, a set-based approach using intervals for design variables (i.e., interaction quantities between subsystems and the total system) is useful. Because in the published approaches, no uncertainty consideration was taken into account for the computation of these intervals, they can possibly have inappropriate size, i.e., being too narrow. The work presented here proposes to include these uncertainties related to design variables. This allows now to consider lack-of-knowledge uncertainty specific for early phase developments in the framework of complex systems design. An example taken from a standard crash load case (frontal impact against a rigid wall) illustrates the proposed methodology.

Shape Optimization Using a Variable Number of Conic Patches

1993 ◽  
Author(s):  
James M. Widmann ◽  
Sheri D. Sheppard
Keyword(s):  
Shape Optimization ◽  
Design Space ◽  
Variable Number ◽  
Optimization Process ◽  
Optimized Design ◽  
Two Dimensional ◽  
Geometric Description ◽  
Design Representation ◽  
Design Variables ◽  
Dimensional Shape

Abstract Shape Optimization is a branch of structural optimization in which the boundaries of geometry are varied. The shape of the boundary is determined by optimizing a set of design variables that form the geometric description of the shape. This paper presents a method of two dimensional shape optimization in which the number of design variables is allowed to change during the optimization process. First an initial design representation is chosen and optimized. Next a new mathematical description of the optimized design is created with an increased number of design variables. This new design is subsequently optimized. This allows the optimization process to work within a larger design space that includes a greater variety of shapes. The process of adding design variables is repeated until no additional improvements in the design are made. Several design examples are solved with this procedure and presented.

Aerodynamic optimization to tandem stators by using sweep and dihedral

2020 ◽  
Vol 234 (6) ◽  
pp. 1225-1236
Author(s):  
Jinouwen Zhang ◽  
Haowan Zhuang ◽  
Jinfang Teng ◽  
Mingmin Zhu ◽  
Xiaoqing Qiang
Keyword(s):  
Gaussian Process ◽  
Surrogate Model ◽  
Aerodynamic Performance ◽  
High Dimensional ◽  
Two Dimensional ◽  
Multi Objective Optimization ◽  
Optimization Scheme ◽  
Multi Objective ◽  
Design Variables ◽  
Dimensional Optimization

In the modern aerodynamic design of turbomachinery blades, the geometries of blades often need to be reshaped to achieve better aerodynamic performance by introducing extra parametric design variables. A higher variable dimension will lead to a larger sampling range as well as a sparser sample distribution, which challenges the effectiveness and stability of optimization schemes based on surrogate model by making the model prediction quality even poorer. In this paper, a multi-objective optimization based on Gaussian process model was carried out for a high dimensional design space. Based on the previous two-dimensional optimization, tandem stators of a modern compressor were optimized by the design of sweep and dihedral. The purpose of the study is to improve the aerodynamic performance of the compressor tandem stators as well as to provide an effective optimization scheme for high dimensional multi-objective optimization problems. The design of sweep and dihedral for reshaping the tandem stators consists of a total of 18 design variables. An improvement in total pressure recovery coefficient of at least 0.7% at positive incidence and at least 0.3% at negative incidence was obtained, much larger than that in the previous two-dimensional optimization. The optimization process shows that, by using Gaussian process as the surrogate model and a special sampling strategy, this optimization scheme is effective and efficient to handle this high dimensional space. The aerodynamic influences of design parameters of tandem blades were analyzed in detail and the superiority of sweep and dihedral in reducing aerodynamic loss was confirmed.

Design Space Analysis Method for Support of System Design Under the Consideration of Uncertainties in the Early Design Stage

2020 ◽  
Author(s):  
Kazuhiro Aoyama ◽  
Yoshihiro Uchibori ◽  
Kazuya Oizumi ◽  
Shigeki Hiramatsu ◽  
Hiroshi Unesaki ◽  
...  
Keyword(s):  
System Design ◽  
Design Space ◽  
Solution Space ◽  
Design Stage ◽  
Design Solution ◽  
Analysis Method ◽  
Early Design Stage ◽  
Design Variables ◽  
Full Picture ◽  
Set Based Design

Abstract In this study, following the concept of set-based design, after preparing global calculation results, we introduced the approach of setting the design solution area that satisfies the product performance goals of the system design. In this approach, from the viewpoint of considering uncertainty, we aimed to develop an analysis method that can get the organic relationship between target variables and design variables. And more, under the assumption that it is difficult to comprehend the full picture of products that are becoming more sophisticated and complex with the knowledge that has been fostered by skilled engineers, the proposed system uses the objective calculation indices that is provided knowledge of the designer. Specifically, the following method are proposed to solve the problem. - Implementation of meta-modeling of design space. - Classified solution space using a density-based clustering method to detect that the solution spaces are divided into multiple disconnected space. - Defined an index called distribution concentration and expressed the possibility of dealing with the uncertainty of the solution domain. - The network diagram based on the calculated index values was proposed to confirm the change in the characteristics of the solution space when the performance target of the product was changed. Finally, the effectiveness of the proposed method was verified by applying it to actual simulation results.

Bayesian Optimization for Multi-Objective High-Dimensional Turbine Aero Design

2021 ◽  
Author(s):  
Yiming Zhang ◽  
Sayan Ghosh ◽  
Thomas Vandeputte ◽  
Liping Wang
Keyword(s):  
Design Space Exploration ◽  
Design Space ◽  
Industrial Applications ◽  
Bayesian Optimization ◽  
High Dimensional ◽  
Multi Objective Optimization ◽  
Loop Optimization ◽  
Multi Objective ◽  
Input Design ◽  
Design Variables

Abstract Industrial design fundamentally relies on high-dimensional multi-objective optimization. Bayesian Optimization (BO) based on Gaussian Processes (GPs) has been shown to be effective for this practice where new designs are picked in each iteration for varying objectives including optimization and model refinement. This paper introduces two industrial applications of BO for turbine aero design. The first application is GE’s Aviation & Power DT4D Turbo Aero Design with 32 design variables. It has a single objective to maximize with 32 input/design variables and thus considered high-dimensional in terms of the input space. BO has significantly succeeded the traditional design schemes. It has been shown that finding the maximum-EI points (inner-loop optimization) could be critical and the influence of inner-loop optimization was evaluated. The second application is for multi-objective optimization. Each simulation run is the aggregate result from multiple CFD runs tuning geometry and took 24 hours to complete. BO has been capable to extend the existing Pareto front with a few additional runs. BO has been searching along the border of the design space and therefore motivate the open-up of design space exploration. For both applications, BO successfully guide the CFD run and allocate design variables more optimum than previous design approaches.

scholarly journals Forest Height Estimation Based on P-Band Pol-InSAR Modeling and Multi-Baseline Inversion

2020 ◽  
Vol 12 (8) ◽  
pp. 1319
Author(s):  
Xiaofan Sun ◽  
Bingnan Wang ◽  
Maosheng Xiang ◽  
Liangjiang Zhou ◽  
Shuai Jiang
Keyword(s):  
Standard Deviation ◽  
Vertical Structure ◽  
Incidence Angle ◽  
Three Dimensional ◽  
Solution Space ◽  
Model Complexity ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Model Inversion ◽  
Forest Height

The Gaussian vertical backscatter (GVB) model has a pivotal role in describing the forest vertical structure more accurately, which is reflected by P-band polarimetric interferometric synthetic aperture radar (Pol-InSAR) with strong penetrability. The model uses a three-dimensional parameter space (forest height, Gaussian mean representing the strongest backscattered power elevation, and the corresponding standard deviation) to interpret the forest vertical structure. This paper establishes a two-dimensional GVB model by simplifying the three-dimensional one. Specifically, the two-dimensional GVB model includes the following three cases: the Gaussian mean is located at the bottom of the canopy, the Gaussian mean is located at the top of the canopy, as well as a constant volume profile. In the first two cases, only the forest height and the Gaussian standard deviation are variable. The above approximation operation generates a two-dimensional volume only coherence solution space on the complex plane. Based on the established two-dimensional GVB model, the three-baseline inversion is achieved without the null ground-to-volume ratio assumption. The proposed method improves the performance by 18.62% compared to the three-baseline Random Volume over Ground (RVoG) model inversion. In particular, in the area where the radar incidence angle is less than 0.6 rad, the proposed method improves the inversion accuracy by 34.71%. It suggests that the two-dimensional GVB model reduces the GVB model complexity while maintaining a strong description ability.

PSS Business Case Map: Supporting Idea Generation in PSS Design

2012 ◽  
Author(s):  
Fumiya Akasaka ◽  
Kazuki Fujita ◽  
Yoshiki Shimomura
Keyword(s):  
Idea Generation ◽  
Business Case ◽  
Literature Survey ◽  
The Self ◽  
High Dimensional ◽  
Self Organizing Map ◽  
Two Dimensional ◽  
Service Type ◽  
Business Cases ◽  
Low Dimensional

This paper proposes the PSS Business Case Map as a tool to support designers’ idea generation in PSS design. The map visualizes the similarities among PSS business cases in a two-dimensional diagram. To make the map, PSS business cases are first collected by conducting, for example, a literature survey. The collected business cases are then classified from multiple aspects that characterize each case such as its product type, service type, target customer, and so on. Based on the results of this classification, the similarities among the cases are calculated and visualized by using the Self-Organizing Map (SOM) technique. A SOM is a type of artificial neural network that is trained using unsupervised learning to produce a low-dimensional (typically two-dimensional) view from high-dimensional data. The visualization result is offered to designers in a form of a two-dimensional map, which is called the PSS Business Case Map. By using the map, designers can figure out the position of their current business and can acquire ideas for the servitization of their business.

Multi-Objective Optimization of an Autonomous Underwater Vehicle

2009 ◽  
Vol 43 (2) ◽  
pp. 48-60 ◽  
Author(s):  
M. Martz ◽  
W. L. Neu
Keyword(s):  
Design Space ◽  
Autonomous Underwater Vehicle ◽  
Underwater Vehicle ◽  
Design Problems ◽  
Single Measure ◽  
Design Variables ◽  
Complex Design ◽  
Final Design ◽  
Undersea Vehicle ◽  
Hierarchy Process

AbstractThe design of complex systems involves a number of choices, the implications of which are interrelated. If these choices are made sequentially, each choice may limit the options available in subsequent choices. Early choices may unknowingly limit the effectiveness of a final design in this way. Only a formal process that considers all possible choices (and combinations of choices) can insure that the best option has been selected. Complex design problems may easily present a number of choices to evaluate that is prohibitive. Modern optimization algorithms attempt to navigate a multidimensional design space in search of an optimal combination of design variables. A design optimization process for an autonomous underwater vehicle is developed using a multiple objective genetic optimization algorithm that searches the design space, evaluating designs based on three measures of performance: cost, effectiveness, and risk. A synthesis model evaluates the characteristics of a design having any chosen combination of design variable values. The effectiveness determined by the synthesis model is based on nine attributes identified in the U.S. Navy’s Unmanned Undersea Vehicle Master Plan and four performance-based attributes calculated by the synthesis model. The analytical hierarchy process is used to synthesize these attributes into a single measure of effectiveness. The genetic algorithm generates a set of Pareto optimal, feasible designs from which a decision maker(s) can choose designs for further analysis.

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