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.

scholarly journals A Globally Optimal Robust Design Method for Complex Systems

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-25
Author(s):  
Yue Chen ◽  
Jian Shi ◽  
Xiao-jian Yi
Keyword(s):  
Complex Systems ◽  
Design Parameter ◽  
Design Method ◽  
Solution Space ◽  
Maximum Size ◽  
Design Criterion ◽  
Design Parameters ◽  
Engineering System ◽  
Output Performance ◽  
Maximum Solution

The uncertainty of the engineering system increases with the growing complexity of the engineering system; therefore, the tolerance to the uncertainty is essential. In the design phase, the output performance should reach the design criterion, even under large variations of design parameters. The tolerance to design parameter variations may be measured by the size of a solution space in which the output performance is guaranteed to deliver the required performance. In order to decouple dimensions, a maximum solution hyperbox, expressed by intervals with respect to each design parameter, is sought. The proposed approach combines the metaheuristic algorithm with the DIRECT algorithm where the former is used to seek the maximum size of hyperbox, and the latter is used as a checking technique that guarantees the obtained hyperbox is indeed a solution hyperbox. There are three advantages of the proposed approach. First, it is a global search and has a considerable high possibility to produce the globally maximum solution hyperbox. Second, it can be used for both analytically known and black-box performance functions. Third, it guarantees that any point selected within the obtained hyperbox satisfies the performance criterion as long as the performance function is continuous. Furthermore, the proposed approach is illustrated by numerical examples and real examples of complex systems. Results show that the proposed approach outperforms the GHZ and CES-IA methods in the literature.

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.

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.

Numerical Optimization of Rotor-Bearing Systems

1989 ◽  
Author(s):  
J. P. Sadler ◽  
K. E. Rouch ◽  
A. S. Rani
Keyword(s):  
Numerical Optimization ◽  
Dynamic Performance ◽  
Optimal Designs ◽  
Design Parameters ◽  
Finite Element Program ◽  
Optimum Placement ◽  
Rotor Bearing ◽  
Element Program ◽  
Programming Techniques ◽  
Selection Of

Abstract Nonlinear programming techniques are combined with a finite element program for dynamic analysis of rotor-bearing systems. The resulting program provides the means for obtaining optimal designs for improved dynamic performance of a rotor through the automated selection of various design parameters of the rotor-bearing system. Both constrained and unconstrained optimizations are considered. Illustrative examples are presented for the case of optimum placement of critical speeds.

scholarly journals Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods

2020 ◽  
Vol 32 (12) ◽  
pp. 2332-2388 ◽  
Author(s):  
Spencer J. Kent ◽  
E. Paxon Frady ◽  
Friedrich T. Sommer ◽  
Bruno A. Olshausen
Keyword(s):  
Hadamard Product ◽  
Solution Space ◽  
Local Information ◽  
Alternative Methods ◽  
High Dimensional ◽  
Factorization Problem ◽  
Gradient Based ◽  
New Type ◽  
Alternative Approaches ◽  
Theoretical Foundations

We develop theoretical foundations of resonator networks, a new type of recurrent neural network introduced in Frady, Kent, Olshausen, and Sommer ( 2020 ), a companion article in this issue, to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a resonator network can efficiently decompose the composite into these factors. We compare the performance of resonator networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that resonator networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and searching in superposition, by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of resonator networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing the guarantee of global convergence, resonator networks are dramatically more effective at finding factorizations than all alternative approaches considered.

scholarly journals Optimization of High-Dimensional Functions through Hypercube Evaluation

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Rahib H. Abiyev ◽  
Mustafa Tunay
Keyword(s):  
Numerical Optimization ◽  
Optimization Problems ◽  
Learning Algorithm ◽  
Evaluation Process ◽  
Stochastic Search ◽  
High Dimensional ◽  
Potential Candidate ◽  
Objective Functions ◽  
Global Numerical Optimization ◽  
Search Area

A novel learning algorithm for solving global numerical optimization problems is proposed. The proposed learning algorithm is intense stochastic search method which is based on evaluation and optimization of a hypercube and is called the hypercube optimization (HO) algorithm. The HO algorithm comprises the initialization and evaluation process, displacement-shrink process, and searching space process. The initialization and evaluation process initializes initial solution and evaluates the solutions in given hypercube. The displacement-shrink process determines displacement and evaluates objective functions using new points, and the search area process determines next hypercube using certain rules and evaluates the new solutions. The algorithms for these processes have been designed and presented in the paper. The designed HO algorithm is tested on specific benchmark functions. The simulations of HO algorithm have been performed for optimization of functions of 1000-, 5000-, or even 10000 dimensions. The comparative simulation results with other approaches demonstrate that the proposed algorithm is a potential candidate for optimization of both low and high dimensional functions.

scholarly journals Hubcap and ignition switch designs - case studies in Independence Axiom

2018 ◽  
Vol 223 ◽  
pp. 01022
Author(s):  
Hilario (Larry) Oh
Keyword(s):  
Case Studies ◽  
Functional Independence ◽  
Design Parameters ◽  
Design Matrix ◽  
Independence Axiom ◽  
Jacobian Determinant ◽  
Mathematical Basis ◽  
Good Design ◽  
Target Values ◽  
Triangular Design

Independence Axiom offers designers a guide to good design. It declares that the design parameters (DPs) conceived for a good design must maintain the independence of the design functional requirements (FRs). Specifically, by relating FRs to DPs through a design matrix [DM] with elements ∂FRi/∂DPj, Independence Axiom declares that only designs with diagonal or triangular design matrix can maintain the functional independence of FRs; and that they should be the only acceptable ones. Starting with the formal definition of functional independence, we derive the criterion for functional independence of FRs as the Jacobian determinant | J | ≠ 0; where the Jacobian matrix [ J ] is shown to be identically equal to [DM]. We further show that if and only if | J | ≠ 0 can the design FRs achieve their target values. Thus the criterion | J | ≠ 0 substantiates the declaration of Independence Axiom since determinant of a diagonal or triangular design matrix is not equal to zero. It serves as the mathematical basis for teaching and implementing Independence Axiom in design. Two case studies are presented to illustrate the implementation of Independence Axiom via the Jacobian determinant | J |.

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