General Solution Methods for Constrained Optimization

2019 ◽  
pp. 173-193
Author(s):  
H. A. Eiselt ◽  
Carl-Louis Sandblom
Keyword(s):  
General Solution ◽  
Constrained Optimization ◽  
Solution Methods

Simulation of Mating Between Non-Analytic Surfaces Using a Mathematical Programming Formulation

2005 ◽  
Author(s):  
Robert Scott Pierce ◽  
David Rosen
Keyword(s):  
Constrained Optimization ◽  
End Milling ◽  
Tolerance Analysis ◽  
Mechanical Assembly ◽  
Solution Algorithms ◽  
Solution Methods ◽  
Gradient Based ◽  
Unconstrained Problem ◽  
Typical Component ◽  
Randomized Search

In this paper we describe a new method for simulating mechanical assembly between components that are composed of surfaces that do not have perfect geometric form. Mating between these imperfect form surfaces is formulated as a constrained optimization problem of the form “minimize the distance from perfect fit, subject to non-interference between components.” We explore the characteristics of this mating problem and investigate the applicability of several potential solution algorithms. The problem can be solved by converting the constrained optimization formulation into an unconstrained problem using a penalty-function approach. We describe the characteristics of this unconstrained formulation and test the use of two different solution methods: a randomized search technique and a gradient-based method. We test the algorithm by simulating mating between component models that exhibit form errors typically generated in end-milling processes. These typical component variants are used as validation problems throughout our work. Results of two different validation problems are presented. Using these results, we evaluate the applicability of the mating algorithm to the problem of mechanical tolerance analysis for assemblies and mechanisms.

Foundation of the Analytic Element Method

2020 ◽  
pp. 71-102
Author(s):  
David R. Steward
Keyword(s):  
General Solution ◽  
Just In Time ◽  
Mathematical Functions ◽  
Analytic Element Method ◽  
Analytic Element ◽  
Solution Methods ◽  
Concise Representation ◽  
Analytic Elements ◽  
Element Method

The Analytic Element Method provides a foundation to solve boundary value problems commonly encountered in engineering and science, where problems are structured around elements to organize mathematical functions and methods. While this text mostly adheres to a ``just in time mathematics'' philosophy, whereby mathematical approaches are introduced when they are first needed, a comprehensive paradigm is presented in Section 2.1 as four steps necessary to achieve solutions. Likewise, Section 2.2 develops general solution methods, and Section 2.3 presents a consistent notation and concise representation to organize analytic elements across the broad range of disciplinary perspectives introduced in Chapter 1.

Einstellung by Observation

1967 ◽  
Vol 21 (2) ◽  
pp. 345-352 ◽  
Author(s):  
Milton E. Rosenbaum ◽  
Sidney J. Arenson
Keyword(s):  
Problem Solving ◽  
General Solution ◽  
Solution Set ◽  
Direct Solution ◽  
Solution Methods ◽  
Method Of Solution

In 2 experiments female Ss observed a confederate solve 6 “water-jar” problems. Each problem permitted 2 modes of solution. In Exp. I the 3 conditions were observation of direct solutions, indirect solutions, or a mixture of both. In subsequent performances on similar problems, observers adopted the solution methods that had been observed even when inefficient. In Exp. II the order of the jar capacities was varied to test for the acquisition by observation of a more general solution set. The 3 conditions were observation of indirect solutions, direct solutions, or no observation (control). The results for both method of solution adopted and time for solution indicate that observation of the direct solution did not facilitate problem solving, but observation of the indirect solution did hinder this activity.

The Use of Transformation and Superposition in the Solution of Multiply-connected Boundary Value Problems

1988 ◽  
Vol 12 (1) ◽  
pp. 1-8
Author(s):  
A. Mioduchowski ◽  
G. Faulkner ◽  
B. Kim
Keyword(s):  
Finite Element ◽  
General Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Boundary ◽  
Multiply Connected ◽  
Solution Methods ◽  
Finite Element Procedure ◽  
Mathematical Techniques ◽  
Linear Boundary Value Problem

The development of two general solution methods for linear multiple boundary value problems was considered in terms of the elastic torsion of multiply-connected inhomogenious prisms. Two mathematical techniques, the transformation and the superposition, are introduced, and, in conjunction with the finite element procedure, they were applied to several example problems for the purpose of verification of the accuracy/reliability. The results obtained indicated that the two methods are versatile, easily adaptable to any arbitrary multiply-connected linear boundary value problem, and provide reliable results.

The Role of VLBI in the Establishment of Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 293-295 ◽  
Author(s):  
I. Zhongolovitch
Keyword(s):  
General Solution ◽  
Future Development ◽  
Rational Approach ◽  
Radio Telescopes ◽  
Coordinate Systems ◽  
Tectonic Plates ◽  
Astronomical Observations ◽  
Optical Instruments ◽  
Fundamental Polyhedron

Considering the future development and general solution of the problem under consideration and also the high precision attainable by astronomical observations, the following procedure may be the most rational approach:1. On the main tectonic plates of the Earth’s crust, powerful movable radio telescopes should be mounted at the same points where standard optical instruments are installed. There should be two stations separated by a distance of about 6 to 8000 kilometers on each plate. Thus, we obtain a fundamental polyhedron embracing the whole Earth with about 10 to 12 apexes, and with its sides represented by VLBI.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

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