Minimum Weight Design Problem Research Articles

Design Sensitivity Analysis of Three-Dimensional Body by Boundary Element Method and Its Application to Shape Optimization.

A design sensitivity calculation technique based on the implicit differentiation method is fomulated for isoparametric boundary elements for three-dimensional(3D) shape optimization problems. The practical sensitivity equations for boundary displacements and stresses are derived, and efficiency and accuracy of the technique are compared with the semi-analytic method by implementing the sensitivity analysis of typical and basic shape design problems numerically. The sensitivity calculation technique is then applied to the minimum weight design problems of 3D bodies under stress constraints, such as the shape optimization of the ellipsoidal cavity in a cube and the connecting rod, where the Taylor series approximation, based on the boundary element sensitivity analysis at current design point, is adopted for the efficient implementation of the optimization.

Synthesis of Torsional Vibration Systems

Abstract Methods of synthesizing discrete torsional vibration systems are developed in this paper. Both minimum weight design (with eigenvalue constraint) and maximum eigenvalue design (with total weight constraint) problems are addressed. Based on the optimality conditions for an optimum design and the equation for the eigenproblem, simple iterative design algorithms are derived. Two numerical examples are included to illustrate the proposed methods.

Efficient Approximation Method in Minimum Weight Design of Stiffened Plate and Sheel Structures. 1st Report, Problems under Displacement and Stress Constraints.

Taylor series approximation (TSA) has been used to form an approximate subproblem to the actual design problem for the efficiency. A new intermediate variable approximation method for stiffened plate and shell structures is suggested and minimum weight design under displacement and stress constraints is discussed. The approximation method of intermediate variables is considered for the thickness and geometric dimensions of the plate or shell and the stiffeners to construct a wide-range approximation model. For the displacement and stress constraints, it is found that the Taylor series approximation using reciprocal variables of thickness and geometric dimensions (RTSA) gives good approximations for the typical stiffened plate structures. The approximation method is also applied to solve the minimum weight design problems of stiffened plate and shell structures, and the results are compared with those of direct variable Taylor series approximation (DTSA) to illustrate the efficiency of optimization.

Optimum design of multiple configuration structures for frequency constraints

Introduction D the service of an aerospace structure, its may vary for different operating conditions. When we restrict the word configuration to structural geometry, the structures with multiple configurations that come to mind are, for example, a tilt-rotor aircraft in airplane or helicopter mode and a stowed vs deployed solar array system for a spacecraft. If we generalize the word to encompass boundary conditions and mass distribution, then all aerospace vehicles can be considered to be structures with multiple configurations. In application, design specifications may impose specific natural frequency constraints for different configurations. For a structure with a single configuration, the minimum weight design with frequency constraints can be found readily.' However, the resulting optimum design may not satisfy frequency constraints for the structure in other configurations. This often leads to design iterations among various configurations. After many design iterations, the final design may be feasible but not optimum. The purpose of this paper is to develop an approach for minimum weight design of structures with several configurations under natural frequency constraints. In the optimum design problem, frequency constraints for all configurations are considered simultaneously. In each design cycle, the constrained minimum weight design problem is solved iteratively by a combined analysis/optimization procedure. To increase the computational efficiency, the iterative analysis is approximated by using reduced-order modal space models.' Sequential linear programming (SLP)'' for optimization procedure is considered. In SLP, the optimum design problem is linearized at each design iteration. Together with a strategy to compute move limits, the resulting linear optimization problem is solved as a linear programming problem. The combination of finite element analysis together with optimization method (SLP) constitutes an effective and reliable approach for solving practical optimum design problems.

Minimum-weight design of structures with natural-frequency constraints

The problem of minimum-weight design of structures with several natural-frequency constraints is considered in this paper. The problem is solved by using a combined finite element method and sequential linear programming (FEM-SLP) formulation. The unique features of the present approach include the use of the assumed mode reanalysis formulation for the repeated eigensolution and the associated sensitivity analysis. The present approach has been implemented using the general-purpose finite element program msc/nastran. Two examples are included to illustrate the effectiveness of the present approach.

Optimization of hybrid aluminum/cfrp box beams

Hybrid structural components in which a principal aluminum structure is reinforced with unidirectional CFRP composite laminates are an effective way to meet the requirements of low weight, high strength and stiffness, relative ease of design and fabrication and good reliability. Designing such components must be done by a procedure appropriate to the material system, with a careful consideration of the failure modes and the possibility of optimizing the component. In this study we analyze the problem of minimum weight design of CFRP-reinforced thin-walled rectangular aluminum sections subjected to given strength and stiffness constraints. The problem is formulated within the framework of classical nonlinear optimization analysis. It is shown that at the optimum design various local failure mechanisms occur simultaneously, while stiffness rarely controls. Experimental results support the accuracy of the calculations and provide additional information on the flexural behavior of hybrid aluminum/CFRP members.

An Efficient Shape Optimization of Minimum Weight Design by Boundary Element Method.

An efficient approximation method to determine an optimum shape of minimum weight subjected to stress constraints is suggested. Design variable reduction techniques of isoparametric interpolation and trigonometric series interpolation for the boundary shape are adopted for reducing the degree of freedom of the design problems. The objective function of weight is approximated to an expansion of a second-order Taylor series and the stress constraints to expansions of first-order Taylor series, based on the boundary element sensitivity analysis at a design point. Then the approximated subproblem is solved by a linear complementary pivot method. In the design variable reduction of isoparametric interpolation, an adaptive mesh refinement technique is also adopted to maintain the accuracy of the structural analysis. The efficiency of the approximation method suggested here is confirmed by application to minimum weight design problems to determine hole shapes in a plate under biaxial loadings.

OPTIMUM DESIGN OF THIN-PLATED STEEL BOX COLUMNS

This paper deals with the optimum design of thin-plated box columns subjected to axial thrust and biaxial end moments. Design formulae based on an effective width approach, proposed by the authors, are utilized in the optimization studies. The problem is first viewed as a strength maximization problem in which the optimum plate slenderness is evaluated for columns having constant weight of material. Results are presented in the form of nondimensional charts so that they can be used for practical design. The problem is then viewed as a minimum weight design problem and solved by means of the SUMT method. Side constraints, such as minimum plate thickness requirements, are imposed during the optimization process to study their effects on the design of box sections. Results are presented in terms of cross-sectional dimensions and minimum weights for columns having different proportions and subjected to different end loadings.

A new implementation of the force method and the optimum design of large trusses

AbstractAn implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system. The matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices. No computational procedure is required to generate the compatibility conditions (or the self‐stress matrix) and no partitioning of the force vector is made into a basic set and a redundant set, unlike the conventional force method. A slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns. This is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions. A sparse generalized reduced gradient package is used as the optimizer. A class of test problems involving large truss structures is solved. The results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures.

An efficient algorithm for the optimum design of trusses with discrete variables

AbstractA method to efficiently solve the problem of minimum weight design of plane and space trusses with discrete or mixed variables is developed. The method can also be applied to continuous variables. The original formulation leads to a non‐linear constrained minimization problem with inequality constraints, which is solved by means of a sequence of approximate problems using dual techniques. In the dual space, the objective function is to be maximized, depends on continuous variables, is concave and has first and second order discontinuities. In addition, the constraints deal simply with restricting the dual variables to be non‐negative. To solve the problem an ad hoc algorithm from mathematical programming has been adapted.Some examples have been developed to show the effectiveness of the method.

SADDLE: A computer-aided structural analysis and dynamic design language—Part I. design system

The SADDLE system is a tool for computer-aided optimal design of structural and mechanical systems. The system is divided into four parts—the preprocessor, the ‘analyzer’, the ‘synthesizer’, and the postprocessor. The structural model and the design data are generated by the preprocessor. The analyzer uses the finite-element method to compute deflections and stresses in the structure. The synthesizer solves the design problem that is cast into a nonlinear programming format. The postprocessor is used to examine the analysis and design results. Different parts of the system interact via a global database. This study presents the design system configuration. In a companion study, the data management part is presented in detail. The user-interface for design is examined first. The design problem formulation, in the nonlinear programming problem context, is presented. The use of utility libraries to achieve this objective with a minimum programming effort is also explained. The article concludes with two numerical examples that belong to two different classes of design problems—a minimum weight design problem and a min-max problem.

Optimum design of beams with different cross-sectional shapes

The minimum weight design problem for frame structures subject to stress and displacement constraints is treated. The cross-sectional dimensions are used as design variables and the hybrid approximation technique in combination with the dual method from mathematical programming is used. Seven different cross-sectional shapes are treated and four examples are solved.

A CAD approach to the minimum weight design problem

AbstractA structure is defined by a set of governing points. These points can be used to define lines, surfaces and solids. A governing point can be connected to a design variable. The values of the design variables are determined in order to minimize the weight of the structure subject to a number of stress constraints. Three simple structures are then designed.

Minimum Weight Design Problems of Fiber-Reinforced Beam Subjected to Uniform Bending

Problems considered here are that of minimizing the weight of beams, which are subjected to a uniform bending moment and reinforced by the fibers distributed in the direction of beam axis. The beam is simplified as a multilaminate structure, of which the fiber volume percent Vfi of each lamina is considered as the design variables. To formulate this design problem the bending theory of multilaminate beam and the law of mixture for the composite material strength are applied. Furthermore, the sequential linear programming and the sequential unconstrianed minimization techniques are used to obtain the design solutions numerically.

A BI-FACTOR ALGORITHM FOR ADJUSTING STRUCTURAL PARAMETERS

The bi-factor algorithm presented in this paper is a development of the single factor optimally criteria algorithm.3-4 The present algorithm does not execute a trial and error procedure, but converges more quickly and stably. Satisfactory results have been attained over a series of minimum weight design problems (including layout optimization) subjected to such constraints as frequency-prohibited-band. structural global buckling load, displacement amplitudes caused by sinusoidal loads, etc.

On dynamic optimization of Timoshenko beam

The present paper discusses the minimum weight design problem for Timoshenko and Euler beams subjected to multi-frequency constraints. Taking the simply-supported symmetric beam as an example, we reveal the abnormal characteristics of optimal Timoshenko beams, i.e., the frequency corresponding to the first symmetric vibration mode could be higher than the frequency of antisymmetric vibration mode if a very thin and high strip is suitably formed at the middle of the beam, and, optimal Timoshenko beams subjected to two different sets of frequency constraints could have the same minimum weight. The above abnormal characteristics demonstrate the need for including maximum cross sectional area constraint in the problem formulation in order to have a well-posed problem.

Optimum Design of Bridge Girders for Electric Overhead Traveling Cranes

The problem of the design of box-type bridge girders for electric overhead traveling cranes is formulated as a minimum weight design problem with inequality constraints. The restrictions placed on the design problem include limitations on the maximum allowable deflections and stresses as well as on the shock absorbing capacity during accidental collision. The overall stability and rigidity considerations are also taken into account. Several load conditions, as per the code specifications, are considered in the design problem. The resulting nonlinear programming problem is solved by using an interior penalty function method. Numerical examples are given to illustrate the effectiveness of the approach. The resulting computer program is used to make a sensitivity analysis of the problem.

On optimization of circular fins with heat generation

The problem of minimizing the maximum temperature of a purely conducting circular fin with heat generation is studied via a variational technique. An exact solution is obtained for an arbitrary heat source distribution and a general non-linear heat conductivity. The problem of minimum weight design of convecting circular fin is then considered. An exact solution for temperature distribution and shape of the optimum fin with constant heat generation is obtained and discussed.

Minimum weight design of axisymmetric sandwich plates

The problem of minimum weight design of laminated circular sandwich plates with axial symmetry is formulated using optimal control theory. A steepest descent algorithm is used to solve the resulting two-point boundary value problem. Inequality constraints on minimum face and core thicknesses, maximum stress levels, and maximum displacement are included. Both statically determinate and statically indeterminate plates are considered. Several examples are presented illustrating the general configuration of minimum-weight designs and the dramatic weight savings attainable.

A note on the maximum number and density of distribution of members in elastic structures of minimum weight under multiple loading conditions

The problem of minimum weight design of elastic structures under multiple loading conditions is considered. It is shown that the problem can be expressed as a search for feasible deflection patterns coupled with repeated searches for structures of minimum weight for given stiffness. The latter is a Linear Programming Problem and implies an upper limit, different from that set by connectivity, on both the number and distribution density of elements present in the minimum weight design.