Optimality Criteria Approach Research Articles

Phase-field method combined with optimality criteria approach for topology optimization

This paper introduces a novel topology optimization framework by combining the phase-field method and optimality criteria approach. This novel approach allows for the automatical addition of holes in the initial shape, a capability that is not achievable with the conventional phase-field model based on boundary changes for topology optimization. The distribution of the material is changed according to the sensitivity of the optimality criteria approach to nucleate holes. After the nucleation of the new holes, they are utilized for topology optimization using the phase-field method. To demonstrate the accuracy and effectiveness of the proposed approach, we conduct several numerical examples based on the two-dimensional minimum compliance problem. The obtained numerical results show the proposed method's fast convergence compared to the conventional phase-field method.

Comparative Study of Peridynamics and Finite Element Method for Practical Modeling of Cracks in Topology Optimization

Recently, topology optimization of structures with cracks becomes an important topic for avoiding manufacturing defects at the design stage. This paper presents a comprehensive comparative study of peridynamics-based topology optimization method (PD-TO) and classical finite element topology optimization approach (FEM-TO) for designing lightweight structures with/without cracks. Peridynamics (PD) is a robust and accurate non-local theory that can overcome various difficulties of classical continuum mechanics for dealing with crack modeling and its propagation analysis. To implement the PD-TO in this study, bond-based approach is coupled with optimality criteria method. This methodology is applicable to topology optimization of structures with any symmetric/asymmetric distribution of cracks under general boundary conditions. For comparison, optimality criteria approach is also employed in the FEM-TO process, and then topology optimization of four different structures with/without cracks are investigated. After that, strain energy and displacement results are compared between PD-TO and FEM-TO methods. For design domain without cracks, it is observed that PD and FEM algorithms provide very close optimum topologies with a negligibly small percent difference in the results. After this validation step, each case study is solved by integrating the cracks in the design domain as well. According to the simulation results, PD-TO always provides a lower strain energy than FEM-TO for optimum topology of cracked structures. In addition, the PD-TO methodology ensures a better design of stiffer supports in the areas of cracks as compared to FEM-TO. Furthermore, in the final case study, an intended crack with a symmetrically designed size and location is embedded in the design domain to minimize the strain energy of optimum topology through PD-TO analysis. It is demonstrated that hot-spot strain/stress regions of the pristine structure are the most effective areas to locate the designed cracks for effective redistribution of strain/stress during topology optimization.

Exact Optimization of a Three-Element Dynamic Vibration Absorber: Minimization of the Maximum Amplitude Magnification Factor

In this study, the maximum amplitude magnification factor for a linear system equipped with a three-element dynamic vibration absorber (DVA) is exactly minimized for a given mass ratio using a numerical approach. The frequency response curve is assumed to have two resonance peaks, and the parameters for the two springs and one viscous damper in the DVA are optimized by minimizing the resonance amplitudes. The three-element model is known to represent the dynamic characteristics of air-damped DVAs. A generalized optimality criteria approach is developed and adopted for the derivation of the simultaneous equations for this design problem. The solution of the simultaneous equations precisely equalizes the heights of the two peaks in the resonance curve and achieves a minimum amplitude magnification factor. The simultaneous equations are solvable using the standard built-in functions of numerical computing software. The performance improvement of the three-element DVA compared to the standard Voigt type is evaluated based on the equivalent mass ratios. This performance evaluation is highly accurate and reliable because of the precise formulation of the optimization problem. Thus, the advantages of the three-element type DVA have been made clearer.

On the influence of local and global stress constraint and filtering radius on the design of hinge-free compliant mechanisms

Distributed compliant mechanisms are components that use elastic strain to obtain a desired kinematic behavior. Compliant mechanisms obtained via topology optimization using the standard approach of minimizing/maximizing the output displacement with a spring at the output port, representing the stiffness of the external medium, usually contain one-node connected hinges. Those hinges are undesired since an ideal compliant mechanism should be a continuous part. This work compares the use of two strategies for stress constrained problems: local and global stress constraints, and analyses their influence in eliminating the one-node connected hinges. Also, the influence of spatial filtering in eliminating the hinges is studied. An Augmented Lagrangian formulation is used to couple the objective function and constraints, and the resulting optimization problem is solved by using an algorithm based on the classical optimality criteria approach. Two compliant mechanisms problems are studied by varying the stress limit and filtering radius. It is observed that a proper combination of filtering radius and stress limit can eliminate one-node connected hinges.

Design and Optimal Synthesis of Compliant Mechanisms - an Optimality Criteria Approach

Design and Optimal Synthesis of Compliant Mechanisms - an Optimality Criteria Approach

二重動吸振器の最大振幅倍率最小化設計

The optimal design method of double-mass dynamic vibration absorbers (DVAs) is discussed with respect to the minimization of the maximum amplitude magnification factor. The performance of a double-mass DVA is superior to a single-mass DVA with the same mass ratio, although the design methods are still the subject of studies. The design optimizations of double-mass DVAs that are arranged in parallel or series are formulated precisely using an optimality criteria approach in which the optimal parameters are obtained as the numerical solutions of simultaneous algebraic equations. The primal equations are derived using Vieta's formula with the assumption that the optimal design is realized with three resonant points of equal height. The additional equations are derived as the determinants of Jacobian matrices that are defined using the primal equations. After rearrangement, these formulations realize the direct numerical solution of the design optimization via the solution of simultaneous algebraic equations. Examples are provided that prove the effects of double-mass DVAs. The formulations used in this study are variants of the algebraic approach developed by O. Nishihara that realized the closed-form algebraic exact solutions to a popular design optimization of a single-mass DVA. The well-known design formulae that use the fixed points by J. P. Den Hartog and J. E. Brock correspond to an approximate solution to this problem.

Optimization Of Curved Plated Structures With The Finite Strip And Finite Element Methods

Abstract The aim of this study is to compare two available numerical tools for solving of partial differential equations for the optimal design of structures. In the past years numerous methods were developed for topology optimization, from these we have adopted the optimality criteria (OC) approach. The main idea is that we state the optimal conditions, that the minimizer has to fulfil at the end of an iterative proves. This method however is not a general one, only advantageous in the case of separable problems, but comes with fast speed, easy programming, and a relative insensitivity of computational time to the number of variables. In the paper we suggest a new method for the elimination of a numerical error, the so called ‘checkerboard pattern’. In the presented examples we applied one loading case and an elastic material behaviour. The cost function is the net weight of the structure and upper bound of the compliance is set as the optimality constraint.

Distributed material density and anisotropy for optimized eigenfrequency of 2D continua

A practical approach to optimize a continuum/structural eigenfrequency is presented, including design of the distribution of material anisotropy. This is often termed free material optimization (FMO). An important aspect is the separation of the overall material distribution from the local design of constitutive matrices, i.e., the design of the local anisotropy. For a finite element (FE) model the amount of element material is determined by a traditional optimality criterion (OC) approach. In this respect the major value of the present formulation is the derivation of simple eigenfrequency gradients with respect to material density and from this values of the element OC. Each factor of this expression has a physical interpretation. Stated alternatively, the optimization problem of material distribution is converted into a problem of determining a design of uniform OC values. The constitutive matrices are described by non-dimensional matrices with unity norms of trace and Frobenius, and thus this part of the optimized design has no influence on the mass distribution. Gradients of eigenfrequency with respect to the components of these non-dimensional constitutive matrices are therefore simplified, and an additional optimization criterion shows that the optimized redesign of anisotropy are described directly by the element strains. The fact that all components of an optimal constitutive matrix are expressed by the components of a strain state, imply a reduced number of independent components of an optimal constitutive matrix. For 3D problems from 21 to 6 parameters, for 2D from 6 to 3 parameters, and for axisymmetric problems from 10 to 4 parameters.

Structural Optimization of Building Materials Using Optimality Criteria Approach and its Realization in ANSYS

In this paper, the structural optimization of building materials is discussed focusing on the explicit formulation of the constraints and the optimal iteration algorithm. The whole optimization procedure was realized by programming using the APDL provided by the commercial software ANSYS. Some key points in programming are discussed such as how to determine a design variable is active or inactive during optimization iteration process. Finally an example was illustrated to demonstrate the validation of the optimization algorithm and the programming method using APDL.

Multiobjective Optimization for Performance-Based Design of Reinforced Concrete Frames

In order to meet the emerging trend of performance-based design of structural systems, attempts have been made to develop a multiobjective optimization technique that incorporates the performance-based seismic design methodology of concrete building structures. Specifically, the life-cycle cost of a reinforced concrete building frame is minimized subject to multiple levels of seismic performance design criteria. In formulating the total life-cycle cost, the initial material cost can be expressed in terms of the design variables, and the expected damage loss can be stated as a function of seismic performance levels and their associated failure probability by the means of a statistical model. Explicit formulation of design constraints involving inelastic drift response performance caused by pushover loading is expressed with the consideration of the occurrence of reinforced concrete plasticity and the formation of plastic hinges. Due to the fact that the initial material cost and the expected damage loss are conflicting by nature, the life-cycle cost of a building structure can be posed as a multiobjective optimization problem and solved by the e-constraint method to produce a Pareto optimal set, from which the best compromise solution can be selected. The methodology for each Pareto optimal solution is fundamentally based on the Optimality Criteria approach. A ten-story planar framework example is presented to illustrate the effectiveness of the proposed optimal design method.

Optimal performance-based design of FRP jackets for seismic retrofit of reinforced concrete frames

External bonding of fiber-reinforced polymer (FRP) composites is now a well-established technique for the strengthening/retrofit of reinforced concrete (RC) structures. In particular, confinement of RC columns with FRP jackets has proven to be very effective in enhancing the strength and ductility of columns, and has become a key technique for the seismic retrofit of RC structures. Despite the large amount of research on the behavior of RC columns confined with FRP, little research has been conducted on the behavior of RC frames with FRP-confined columns. For the seismic retrofit of RC frames with FRP, apart from the structural response of a retrofitted frame, an important issue is how to deploy the least amount of the FRP material to achieve the required upgrade in seismic performance. With these two issues in mind, this paper presents an optimization technique for the performance-based seismic FRP retrofit design of RC building frames. The thicknesses of FRP jackets used for the confinement of columns are taken as the design variables, and minimizing the volume and hence the material cost of the FRP jackets is the design objective in the optimization procedure. The pushover drift is expressed explicitly in terms of the FRP sizing variables using the principle of virtual work and the Taylor series approximation. The optimality criteria (OC) approach is employed for finding the solution of the nonlinear seismic drift design problem. A numerical example is presented and discussed to demonstrate the effectiveness of the proposed procedure.

Nonlinear Stiffness Design Optimization of Tall Reinforced Concrete Buildings under Service Loads

Tall reinforced concrete (RC) building designs must satisfy serviceability stiffness criteria in terms of maximum lateral displacement and interstory drift. It is therefore important to assess accurately the effects of concrete cracking on lateral stiffness of such structures. This study integrates nonlinear cracking analysis methods with a powerful optimization technique and presents an effective numerical approach for the stiffness-based optimum design of tall RC buildings under service loads. A probability-based effective stiffness method is employed to identify cracked members and to modify their effective cracked stiffness. Iterative procedures are necessary for the serviceability analysis of tall RC buildings to determine their nonlinear stiffness characteristics due to concrete cracking. Design optimization based on a rigorously derived optimality criteria approach involves minimizing the cost of RC structures while satisfying the top and multiple interstory drift constraints along with member sizing requirements. A framework example is presented to illustrate the applicability and efficiency of this proposed optimal design tool. Discussions about the effects of concrete cracking on the optimization results are also included.

A new optimality criteria method for shape optimization of natural frequency problems

A new shape optimization method for natural frequency problems is presented. The approach is based on an optimality criterion for general continuum solids, which is derived in this paper for the maximization of the first natural frequency with a volume constraint. An efficient redesign rule for frequency problems is developed to achieve the required shape modifications. The optimality criterion is extended to volume minimization problems with multiple frequency constraints. The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization. The combination with the optimality criteria approach results in a robust and fast convergence, which is independent of the number of design variables. Sensitivity information of objective function and constraints are not required, which allows to solve the structural analysis task using fast and reliable industry standard finite element solvers like ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN, or PERMAS. The new approach is currently being implemented in the optimization system TOSCA.

A fuzzy proportional-derivative controller for engineering optimization problems using an optimality criteria approach

This paper proposes a fuzzy proportional-derivative (PD) controller optimization engine for engineering optimization problems using an optimality criteria approach. Traditional numerical optimization algorithms treat optimization problems as pure mathematical problems. Engineering knowledge about the problem is not utilized in the optimization process. The idea of using the fuzzy PD controller in engineering optimization is that, instead of using purely numerical information to obtain the new design point in the next iteration, engineering knowledge and human supervision process can be modeled in the optimization algorithm using fuzzy rules. The fuzzy PD controller optimization engine developed in this work appears to have stable performance in both structural optimization and blow molding parameter optimization examples.

Elastic and inelastic drift performance optimization for reinforced concrete buildings under earthquake loads

AbstractThis paper presents an effective optimization technique for the elastic and inelastic drift performance design of reinforced concrete buildings under response spectrum loading and pushover loading. Attempts have been made to develop an automatic optimal elastic and inelastic drift design of concrete framework structures. The entire optimization procedure can be divided into elastic design optimization and inelastic design optimization. Using the principle of virtual work, the elastic drift response generated by the response spectrum loading and the inelastic drift response produced by the non‐linear pushover loading can be explicitly expressed in terms of element sizing design variables. The optimization methodology for the solution of the explicit design problem of buildings is fundamentally based on the Optimality Criteria approach. One ten‐story, two‐bay building frame example is presented to illustrate the effectiveness and practicality of the proposed optimal design method. While rapid convergence in a few design cycles is found in the elastic optimization process, relatively slow but steady and smooth convergence of the optimal performance‐based design is found in the inelastic optimization process. Copyright © 2004 John Wiley & Sons, Ltd.

Structural Design Optimization of Nonlinear Symmetric Structures Using the Group Theoretic Approach

Among multidisciplinary analysis and optimization problems, structural optimization of geometrically nonlinear stability problems is of great importance, especially in structures used in space applications, because of their long and slender configurations. In this study, application of the group theoretic approach (GTA) in structural optimization of geometrical nonlinear problems under system stability constraint has been investigated. According to GTA, the number of displacement degrees of freedom in the initial configuration can be reduced significantly by using the set of symmetry transformations of the undeformed structure to construct a projection matrix from full space to a reduced subspace spanned by the symmetry modes. A structural optimization algorithm is developed for shallow structures undergoing large deflections subject to system stability constraint. The method combines the nonlinear buckling analysis, based on the displacement control technique using GTA, with the optimality criteria approach, based on the potential energy of the system. A shallow dome truss structure has been designed to illustrate the proposed methodology. This paper demonstrates that structural optimization of nonlinear symmetric structures using GTA is computationally efficient, and excellent agreement exists between optimal results in full space and those in the reduced subspace. Also, it is shown that structural design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass, which may lead to structural failure.

Optimum design of truss structures undergoing large deflections subject to a system stability constraint

A structural optimization algorithm is developed for shallow trusses undergoing large deflections subject to a system stability constraint. The method combines the non-linear buckling analysis, through displacement control technique, with the optimality criteria approach. Four examples illustrate the procedure and allow the results obtained to be compared with those in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass for these types of structures so a design based on the linear buckling analysis can result in catastrophic failure. In one of the design examples the stresses in the elements, in the optimum design, exceed the allowable stresses, pointing out the need for a design that accounts for both non-linear buckling and stress constraints. Copyright © 2000 John Wiley & Sons, Ltd.

On optimal shapes in materials and structures

In the micromechanics design of materials, as well as in the design of structural connections, the boundary shape plays an important role. The objective may be the stiffest design, the strongest design or just a design of uniform energy density along the shape. In an energy formulation it is proven that these three objectives have the same solution, at least within the limits of geometrical constraints, including the parametrization. Without involving stress/strain fields, the proof holds for 3D-problems, for power-law nonlinear elasticity and for anisotropic elasticity.¶To clarify the importance of parametrization, the problem of material/hole design for maximum bulk modulus is analysed. A simple optimality criterion is derived and with a simple superelliptic parametrization, agreement with Hashin-Shtrikman bounds are found. More general examples including nonequal principal strains, nonlinear elasticity and orthotropic elasticity show the versatility of the optimality criterion approach. In spite of this, the mathematical programming approach will be used in the future study of the multiparameter and/or multipurpose problems.

Optimum design of unbraced rigid frames

In this paper, an algorithm is presented for the optimum design of unbraced rigid frames which takes into account the non-linear response of the frame due to the effect of axial loads. It considers sway constraints and combined stress limitations in the design problem. While the optimality criteria approach is employed to handle sway limitations, the combined stress constraints are reduced to non-linear equations of the design variables. The algorithm initiates the design process by carrying out non-linear analysis of the frame. It checks the overall stability in every iteration of the analysis. When the non-linear response of the frame is obtained without loss of stability, the new values of design variables are computed from the recursive relationships. This process of re-analysis and re-sizing is repeated until the convergence is obtained. A number of tall steel frames are designed by the algorithm as examples.

Optimum design of nonlinear elastic framed domes

In this paper, an algorithm is presented for the optimum design of three-dimensional rigidly jointed frames which takes into account the nonlinear response due to the effect of axial forces in members. The stability functions for three-dimensional beam-columns are used to obtain the nonlinear response of the frame. These functions are derived by considering the effect of axial force on flexural stiffness and effect of flexure on axial stiffness. The optimum design algorithm considers displacement limitations and restricts combined stresses not to be more than yield stress. It employs the optimality criteria approach together with nonlinear overall stiffness matrix to develop a recursive relationship for design variables in the case of dominant displacement constraints. The combined stress constraints are reduced into nonlinear equations of design variables. The algorithm initiates the optimum design at the selected load factor and carries out elastic instability analysis until the ultimate load factor is reached. During these iterations checks of the overall stability of frame is conducted. If the nonlinear response is obtained without loss of stability, the algorithm then proceeds to the next design cycle. The method developed is applied to the optimum design of a number of rigid space frames to demonstrate its versatility.