An Optimality Criteria Approach for the Topology Synthesis of Compliant Mechanisms

1998 ◽  
Author(s):  
A. Saxena ◽  
G. K. Ananthasuresh
Keyword(s):  
Energy Density ◽  
Mathematical Programming ◽  
Objective Function ◽  
Compliant Mechanisms ◽  
Optimality Criteria ◽  
Optimality Criteria Method ◽  
New Energy ◽  
Method Of Solution ◽  
Design Parameterization ◽  
The Continuum

Abstract The physical insight used in formulating a multi-criteria optimization problem for the synthesis of compliant mechanisms, is quickly lost if mathematical programming techniques (SLP, SQP etc.) are used to determine the optimal solution. As opposed to the previous works that relied upon mathematical programming search techniques to find the optimum solution, in this paper we present an alternative method of solution called the optimality criteria method. Optimality criteria methods have proven to be effective in structural optimization problems with a large number of variables, and very few constraints as is the case in the topology synthesis of compliant mechanisms. The important new results of this paper include: (i) the derivation of a physically insightful optimal property of compliant mechanisms which states that the ratio of the mutual potential energy density and the strain energy density is uniform throughout the continuum (ii) the development of the optimality criteria method of solution in the form of a simple update formula for the design variables by using the above property (iii) design parameterization using the frame finite-element based ground-structure that appropriately accounts for the requisite bending behavior in the continuum, and (iv) numerical implementation of previously reported density based design parameterization using bilinear plane-stress elements. In addition, a new energy based multi-criteria objective function is presented to maximize the useful output energy (which is equivalent to maximizing the mechanical advantage) while meeting the kinematic requirements. Several examples are included to demonstrate the validity of the optimal property, the optimality-criteria method of solution, and the improvements made possible by the new energy based objective function.

Towards Uniformly Distributed Compliance in Compliant Mechanisms: A Multi-Objective Approach

2006 ◽  
Author(s):  
Stephen L. Canfield ◽  
Daniel L. Chlarson ◽  
Alexander Shibakov ◽  
Patrick V. Hull
Keyword(s):  
Finite Element ◽  
Objective Function ◽  
Compliant Mechanisms ◽  
Optimality Criteria ◽  
Optimal Designs ◽  
Limiting Factors ◽  
Objective Functions ◽  
Multi Objective ◽  
Simple Addition ◽  
Current Formulation

Researchers in the field of optimal synthesis of compliant mechanisms have been working to develop tools that yield distributed compliant devices to perform specific tasks. However, it has been demonstrated in the literature that much of this work has resulted in mechanisms that localize compliance rather than distribute it as desired. In fact, Yin and Ananthasuresh (2003) [1] demonstrate that based on the current formulation of optimality criteria and analysis via the finite element (FE) technique, a lumped compliant device will always exist as the minimizing solution to the objective function. The addition of constraints on allowable strain simply moves the solution back from this objective. Therefore, modification to the standard optimality criteria needs to take place. Yin and Ananthasuresh [1] proposed and compared several approaches that include distributivity-based measures within the optimality criteria, and demonstrated the effectiveness of this approach. In this paper, the authors propose to build on this problem. In a similar manner, a general approach to the topology synthesis problem will be suggested to yield mechanisms in which the compliance is distributed throughout the device. This work will be based on the idea of including compliance distribution directly within the objective functions, while addressing some of the potential limiting factors in past approaches. The technique will be generalized to allow simple addition of criteria in the future, and to deliver optimal designs through to manufacture. This work will first revisit and propose several quantitative definitions for distributed compliant devices. Then, a multi-objective formulation based on a non-dominating sort and Pareto set method will be incorporated that will provide information on the nature of the problem and compatibility of employed objective functions.

Use of Penalty Function in Topological Synthesis and Optimization of Strain Energy Density of Compliant Mechanisms

1997 ◽  
Author(s):  
Mary I. Frecker ◽  
Sridhar Kota ◽  
Noboru Kikuchi
Keyword(s):  
Energy Density ◽  
Penalty Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Compliant Mechanisms ◽  
High Stress ◽  
Optimal Solution ◽  
Optimality Criteria ◽  
Second Stage ◽  
Starting Point

Abstract A penalty function approach is used in conjunction with a multi-criteria optimization method for topology synthesis of compliant mechanisms. This method can help facilitate convergence to physically meaningful solutions for problems with a large number of design variables. The second part of the paper is an investigation of the element strain energy density of the optimal solution, where a second stage size optimization routine is developed. The solution from the topology optimization is used as a starting point for the resizing algorithm, which uses an optimality criteria method based on the overall average strain energy density. This second stage optimization more uniformly distributes the element strain energy densities in order to avoid localized areas of high stress or strain.

PennSyn: A Topology Synthesis Software for Compliant Mechanisms

2000 ◽  
Author(s):  
A. Saxena ◽  
X. Wang ◽  
G. K. Ananthasuresh
Keyword(s):  
Compliant Mechanisms ◽  
Optimality Criteria ◽  
Detection Algorithm ◽  
Analysis Software ◽  
Boundary Extraction ◽  
Modeling And Analysis ◽  
Software Packages ◽  
Optimality Criteria Method ◽  
Quasi Newton ◽  
Ground Structures

Abstract In this paper, we present PennSyn, a software with graphical user interface that provides an automated design route from function specifications to fabrication of fully compliant mechanisms. PennSyn uses the notion of maximizing the flexibility and stiffness of a continuum simultaneously through multi-criteria objectives for the topology synthesis of compliant mechanisms. Both the optimality criteria and quasi Newton mathematical programming methods are employed as optimization algorithms in PennSyn. Reliability in the optimality criteria method is ensured using a one variable search while the volume constraint is addressed by global resizing of variables without altering the function value locally. The design continuum is represented using ground structures of linear truss and frame finite elements. These element types are easy and robust in implementation and help provide effortless extraction and transfer of optimal topologies into commercial CAD packages. An edge detection algorithm is used with PennSyn for boundary extraction of optimal compliant geometries. The resulting data is stored in IGES format for easy portability into commercial modeling and analysis software packages.

Design Space Analysis of Distributed Compliance in Segmented Beam Templates of Compliant Mechanisms

2009 ◽  
Author(s):  
Stephen L. Canfield ◽  
Alexander Shibakov ◽  
Joseph D. Richardson
Keyword(s):  
Objective Function ◽  
Design Space ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Multiple Objectives ◽  
Elastic Response ◽  
Beam Model ◽  
Closed Form Solutions ◽  
Parametric Studies ◽  
Design Parameterization

A significant amount of research has been conducted in developing optimal synthesis techniques for compliant mechanisms with the expectation that distributed devices would result from the continuum design domain. However, it is commonly noted that much of this work has resulted in mechanisms that show localized rather than distributed compliance. This behavior has been attributed to a variety of sources including numerical discrepancies in the model, objective function formulation, and design parameterizations. In this paper, the nature of compliance distribution over particular objective function formulations and design parameterization are further considered in the absence of numerical or resolution issues. The intent is to better understand the behavior of the objective function over multidimensional subsets of the design space that include a direct measure for distribution of compliance. The approach is based on a simple, representative compliant mechanism formed as a segmented beam model. This mechanism is considered to be representative of compliant mechanism behavior in systems where elastic deformation is dominated by bending. Closed-form solutions for the elastic response of this representative mechanism are presented and parametric studies of the response of traditional objectives over subsets of the design space are conducted. The results show that in the absence of numeric artifacts, mechanism efficiencies are improved as mechanisms tend toward lumped compliance when single objectives are considered on mechanisms dominated by bending. However, when more than one objective is deemed important in the design, there exist preferred regions of the workspace, not necessarily in a lumped region, that depend largely on the interaction of the multiple objectives. Of these preferred regions, one lies in a moderately lumped region (h2/h1 ≈ 0.2) and one in a distributed region (h2/h1 ≈ 0.7). The designs in these regions reveal a higher viability in simultaneously satisfying the multiple objectives. This result is based on a visualization of the design space based on measuring the correlation of a multiple objectives over the design space. The results demonstrate several of the factors which contribute to this behavior, and provide an initial measure of the importance of each. Finally, suggestions are provided based on these results that can be used to improve the optimization process if the desire is to achieve distributed compliance.

scholarly journals Fresa: A Plant Propagation Algorithm for Black-Box Single and Multiple Objective Optimization

2021 ◽  
Vol 2 (4) ◽  
Keyword(s):  
Mathematical Programming ◽  
Objective Function ◽  
Optimization Problems ◽  
Population Based ◽  
Black Box ◽  
Multiple Objective ◽  
Multiple Objective Optimization ◽  
Plant Propagation ◽  
Propagation Algorithm ◽  
Multiple Dispatch

Fresa implements a nature inspired plant propagation algorithm for the solution of single and multiple objective optimization problems. The method is population based and evolutionary. Treating the objective function as a black box, the implementation is able to solve problems exhibiting behaviour that is challenging for mathematical programming methods. Fresa is easily adapted to new problems which may benefit from bespoke representations of solutions by taking advantage of the dynamic typing and multiple dispatch capabilities of the Julia language. Further, the support for threads in Julia enables an efficient implementation on multi-core computers.

scholarly journals MATHEMATICAL PROGRAMMING APPLICATIONS PECULIARITIES IN SHAKEDOWN PROBLEM/MATEMATINIO PROGRAMAVIMO TAIKYMO KONSTRUKCIJŲ PRISITAIKYMO UŽDAVINIUOSE YPATUMAI

2001 ◽  
Vol 7 (2) ◽  
pp. 106-114
Author(s):  
Ela Chraptovič ◽  
Juozas Atkočiūnas
Keyword(s):  
Mathematical Programming ◽  
Optimization Problems ◽  
Optimality Criteria ◽  
Point Of View ◽  
Design Matrix ◽  
Problem Solution ◽  
Strain Compatibility ◽  
Complete Set ◽  
Mechanical Interpretation

The theory of mathematical programming widely spread as a method of a solution of extreme problems. It accompanies the study of plastic theory problem from its posing up to final solution. However, here again from our point of view not all possibilities are realized. Unfortunately, the use of mathematical programming as an instrument of a numerical solution for structural analysis frequently is also restricted by that. The possibilities of mechanical interpretation of optimality criteria of applied algorithms are not uncovered. The global solution of the problem of mathematical programming exists, if Kuhn-Tucker conditions are satisfied. These conditions do not depend on the applied algorithm of a problem solution. The identity of Kuhn-Tucker conditions with a optimality criteria of Rosen algorithm is finding out in this research. The role of a design matrix for the creating of strain compatibility equations is clarified. The Kuhn-Tucker conditions mean the residual strain compatibility equations in analysis of elastic-plastic systems. It is proved in the article that for problems of limiting equilibrium the Kuhn-Tucker conditions include the dependences of the associated law of plastic flow. The Kuhn-Tucker conditions together with limitations of a source problem of account represent a complete set of dependences of the theory of shakedown. The correct mathematical and mechanical interpretation of the Kuhn-Tucker conditions allows to refuse a direct solution of a dual problem of mathematical programming. It makes easier the solution of optimization problems of structures at shakedown.

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