Thermomechanical coupling in folded plates and non-smooth shells

In this work we present the model for thermomechanical coupling in folded plates or non-smooth shells, which can be used for analysis of fire-resistance of cellular structures. For this purpose, a modified version of shell element including drilling degrees of freedom permits a detailed modeling combined with a constitutive model proposed in terms of the Saint-Venant multi-surface plasticity criterion. Thermomechanical coupling is also considered including radiative exchanges and operator split solution procedure with different time steps for each sub-problem. The model performance is illustrated by several numerical examples including fire-resistance of walls built with clay bricks.     

Structural optimization of finite deformation elastoplasticity using continuum-based shape design sensitivity formulation

A continuum-based shape design sensitivity formulation and optimization method is proposed for finite deformation elastoplasticity. In response analysis, the multiplicative decomposition of the deformation gradient into elastic and plastic parts is used for the hyperelasticity-based elastoplastic constitutive model with respect to the intermediate configuration. In design sensitivity analysis, the shape variation at the undeformed configuration is taken using a design velocity concept and then is transformed to the current configuration to recover the updated Lagrangian formulation. The design sensitivity equation of the direct differentiation method is solved at each time step without iteration. The effect of using different reference frames for response analysis and sensitivity analysis is discussed in detail. The path-dependency of the sensitivity is due to the evolutions of the intermediate configuration and the internal plastic variables. A numerical example is shown to confirm the accuracy and efficiency of the proposed computational method using a vehicle bumper optimization.     

Structural shape optimization using self-adjusted convex approximation

This study researches the applications of Self-Adjusted Convex Approximation (SACA) in structural shape optimization problems. The B-spline curve is adopted as the mathematical representation of the structural shapes. The SACA method is based on the CONvex LINearization (CONLIN) method and has better accuracy and convergent rate. Numerical examples are offered and the results show that the proposed method is effective in the structural shape design.     

Free material optimization for laminated plates and shells

Free Material Optimization (FMO) is a powerful approach for conceptual optimal design of composite structures. The design variable in FMO is the entire elastic material tensor which is allowed to vary almost freely over the design domain. The imposed requirements on the tensor are that it is symmetric and positive semidefinite. Most of today’s studies on FMO focus on models for two- and three-dimensional structures. The objective of this article is to extend existing FMO models and methods to laminated plate and shell structures, which are used in many engineering applications. In FMO, the resulting optimization problem is generally a non convex semidefinite program with many matrix inequalities which requires special-purpose optimization methods. The FMO problems are efficiently solved by a primal-dual interior point method developed and implemented by the authors. The quality of the proposed FMO models and the method are supported by several large-scale numerical experiments.     

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.     

Modeling of shape memory alloy shells for design optimization

A three-dimensional phenomenological constitutive model for the analysis and design optimization of shape memory alloy (SMA) structures is presented. This model specifically targets the pseudoelastic behavior due to the R-phase transformation in NiTi alloys, but also applies to similar SMA materials with low hysteresis. A history-independent formulation is presented, which allows cost-effective sensitivity analysis. The possibility to efficiently compute design sensitivities is essential for enabling the use of gradient-based optimization algorithms, which will allow design optimization of complex SMA structures. The use of the constitutive model in a problem of realistic complexity is illustrated by the analysis of a SMA miniature gripper, modeled using shell elements. The sensitivity analysis of SMA structures using the presented model is addressed in an accompanying paper.     

Coons' surface method for formulation of finite element of plates and shells

In this paper, we apply the Coons' surface method and fitted function interpolation to fit boundary conditions of the finite element, and obtain different displacement functions of the plate and shell rectangular element, as well as the sectorial element, parallelogram element, ring element and quadrilateral element. The method is easily implemented and its geometric significance and mechanical conception are quite clear. Both computing technique and operational procedure are relatively unified. It is convenient to formulate conforming elements and high-precision elements, and also may be applied to formulate mixed and hybrid elements.     

A hybrid strain technique for finite element analysis of plates and shells

A specialization of the Hu-Washizu [1] functional wherein strains and displacements are taken as independent variables is employed in the formulation of ‘hybrid’ elements. Both the strains and displacements are independently interpolated with the strains being eliminated at the element level, leaving displacement variables only to be assembled into the global system of equations. This distinguishes such elements as ‘hybrid’, in contrast to ‘mixed’ wherein the global system of equations contains all the discretized variables. Applications including ‘thick’ plate and shell elements are considered. In many applications the hybrid strain technique appears more natural than the hybrid stress technique since stress discontinuities are accommodated quite conveniently.     

On the modelling of ribbed plates for shape optimization

The development of a homogenized plate model suitable for shape optimization is presented. The development is based on a homogenization method for layered materials with a periodic microstructure. A particular advantage of the approach is that it accommodates without significant changes the modelling of ribbed, honeycomb and perforated plates. The model is compared with others that have appeared in the literature and that are also useful in the context of the optimization of the shape and layout of plates and plate-like structures. The results indicate that the model presented here is useful in the optimization of both thick and thin plates.     

Structural modelling and sensitivity analysis of shape optimization

It is presented in this paper that the structural modelling of shape optimization is composed of, in general cases, four distinct processes on geometry, design, analysis and perturbation models. The relationships between these models are discussed. An integrated modelling approach based on geometric shape parameterization and automatic mesh generation is proposed. In cooperation with this modelling approach, the semi-analytic sensitivity analysis has been effectively employed. These techniques join shape optimization with FEM and CAD packages and apply it versatilely to optimum designs of general structures. The implementation and applications of the integrated modelling approach and semi-analytic sensitivity analysis to shape optimization of structures with coupling of stress and temperature fields are illustrated.Presented at NATO ASI Optimization of Large Structural Systems, held in Berchtesgaden, Germany, Sept. 23 — Oct. 4, 1991     

Nonlinear finite element analysis of reinforced concrete plates and shells under monotonic loading

Plane stress constitutive models are proposed for the nonlinear finite element analysis of reinforced concrete structures under monotonic loading. An elastic strain hardening plastic stress-strain relationship with a nonassociated flow rule is used to model concrete in the compression dominating region and an elastic brittle fracture behavior is assumed for concrete in the tension dominating area. After cracking takes place, the smeared cracked approach together with the rotating crack concept is employed. The steel is modeled by an idealized bilinear curve identical in tension and compressions. Via a layered approach, these material models are further extended to model the flexural behavior of reinforced concrete plates and shells. These material models have been tested against experimental data and good agreement has been obtained.     

Two dimensional shape optimization using partial control and finite element method for compressible flows

The objective of this study is to determine the two dimensional shape of a body located in a compressible viscous flow, where the applied fluid force is minimized. The formulation to obtain the optimal shape is based on an optimal control theory. An optimal state is defined as a state, in which the performance function defined as the integration of the square sum of the applied fluid forces is minimized due to a reduction in the applied fluid forces. Compressible Navier–Stokes equations are treated as constraint equations. In other words, the body is considered to have a shape that minimizes the fluid forces under the constraint of the Navier–Stokes equations. The gradient of the performance function is computed using the adjoint variables. A weighted gradient method is used as the minimization algorithm. The volume of the body is assumed to be the same as that of the initial body. In the case of the algorithm used in this study, both the creation of a structured mesh around the surface of the body and the smoothing procedure are employed for the computation of gradient. In this study, a remeshing technique based on the structured mesh around the body changing its configuration in the iteration cycle is employed. For the correction to keep the volume constant, the surface coordinates are moved along the radial direction. For the discretization of both the state and adjoint equations, the efficient bubble function interpolation presented previously by the authors [18] is employed. The algorithm, which is known as the partial control algorithm, is applied to the numerical procedure to determine the movement of the coordinates. In the case of the gradient method, in order to avoid the convergence of the final shape to the local minimum shape, the new algorithm, which is called the partial control algorithm, is presented in this study. In numerical studies, the shape determination of a body in a uniform flow field is carried out in 2D domains. The initial shape of the body is assumed to be an elliptical cylinder. The shape is modified by minimizing the applied fluid forces. Finally, the desired shape of a body, whose performance function is reduced and converged to a constant value, is obtained. By carrying out a procedure that involves the use of the partial control algorithm, the desired shape of a body, whose performance function is reduced further, is obtained. Stable shape determination of a body in a compressible viscous flow is carried out by using the presented method. It is indicated that the optimal shape can be obtained by using the partial control algorithm.     

Study of a nine-node mixed formulation finite element for thin plates and shells

A nine-node element, designated as SHEL9, has been developed for analysis of thin plates and shells. The element formulation is based on the degenerate solid shell concept and a modified Hellinger-Reissner principle with independent in-plane and transverse shear strains. Numerical tests indicate that the present SHEL9 element with uniform 3 × 3 point integration rule is free of locking, and it gives reliable solutions even for thin plates and shells.     

Optimum design of plates and shells using the DCOC method

The DCOC method developed recently by Zhou and Rozvany (1992, 1993) has been shown to highly improve the efficiency of optimality criteria methods. In the present paper, the application of this method to plates and shells is discussed. High efficiency of DCOC is guaranteed by the fact that the computational expense of DCOC is only influenced by the number of active displacement constraints, which is usually very small for the considered problem.     

Structural shape optimization integrated with CAD environment

The research work presented here is based on the concept of the integration of optimization techniques and numerical analysis with the finite element method (FEM) and computer-aided design (CAD). A microcomputer aided optimum design system, MCADS, has been developed for general structures. Certain techniques to be discussed in the paper, e.g. the semi-analytical method for design sensitivity analysis, optimization analysis modelling for shape design, application oriented user interfaces and the coupling of automated optimization and user intervention have rendered MCADS pratical and versatile in applications for engineering structures. The above techniques and an application are presented in this paper.     

Maximum energy dissipation for elasto-plastic plates via isogeometric shape optimization

In this research, optimum shape of plate structures is sought to maximize the energy dissipation via structural shape optimization. To achieve this, isogeometric analysis (IGA) is utilized for structural analysis of plates considering elasto-plastic behavior of materials. The von Mises material model is employed for this purpose. Non-uniform rational B-splines basis functions are used for both geometry definition and approximating the unknown deformation field. The optimization problem is to maximize the structural dissipated energy until a prescribed displacement is reached and a fixed amount of material is considered in the design domain. A direct shape sensitivity analysis is performed and a mathematical based approach is employed for the optimization process. To demonstrate the efficiency of the proposed algorithm three examples are illustrated. Using the IGA prevents adjusting analysis model during the optimization process, which is time-consuming especially when iterative nonlinear analysis is performed. The results also show that large geometry modifications can be properly managed by the proposed algorithm.

     

Design of shell shape using finite elements

A finite triangular facet element for the analysis of doubly curved thin shells is presented, the principal feature of which is a particularly simple resolution process. A simple iterative design procedure is developed, the optimality criteria of which are the elimination of bending and the minimization of the surface integral of the membrane stresses. The procedure is used to obtain numerical predictions of the optimal shapes for constant thickness arches and shells that are in good agreement with those expected. Finally, the procedure is extended to provide an optimal shape for a uniform thickness arch dam and an iterative procedure used to provide an optimal thickness variation.