Shape sensitivity analysis of thin-shell structures

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

This paper presents the implementation of the boundary element method to shape sensitivity analysis of elastic structures with stress concentrators. An elastic body which contains a number of voids (internal boundaries), playing the role of stress concentrators, is considered. We are interested in calculating the first order sensitivity of shape–dependent functionals with respect to the shape variation of the body domain. This task is accomplished using the adjoint variable method. As it has been shown by Dems and Mróz [10], for a basic transformation (i.e. a translation, a rotation or a scale change) of the body, the sensitivity of the considered functional takes the form of a path-independent integral (PII) whose integrand depends on the primary and adjoint state fields, along an arbitrary path (curve in 2D, surface in 3D), enclosing the transformed stress concentrator (void). It is very important for numerical computations, because we can compute this integral along the path placed far from the stress concentrator to eliminate the negative influence of stress concentrations on the accuracy of calculations. The boundary element method (BEM) is used to solve both primary and adjoint problems. Some important cases of adjoint problems related to functionals are analyzed in the paper. A thorough numerical verification of the proposed method is performed in this work. The presented method of sensitivity analysis is utilised in gradient-based optimization and identification problems. Numerical examples of optimization and identification are shown in this paper.

Adjoint design sensitivity analysis and optimization of nonlinear structures using geometrical mapping approach

In the previous paper the rigid flat plate element was derived for the discrete limit analysis of thin shell structures of arbitrary shape, and numerical results under the assumption of small deformations were shown on the standard elasto-plastic problems. In general the effect of geometrical nonlinearity such as large deformations and buckling behaviors is one of the main factors which determine the ultimate strength of thin-walled structures, however, the method for the general treatment of geometrical nonlinearity is not necessarily given in the conventional plastic analysis.In the present report the general computational algorithm for the discrete limit analysis of thin-walled plate and shell structures is presented, by using the flat rigid plate element of triangular shape. The present algorithm is characterized by the following statements :(1) The discrete analysis on the limit strength is possible for arbitrary thin-walled structural members, in which the effects of geometrical as well as material nonlinearities are taken into account.(2) In the application to the structural stability problems a family of simulation models with low degrees of freedom can be derived, including Shanley model in the plastic buckling problem of columns and Yoshimura buckle pattern observed in the non-axisymmetric buckling behavior of circular cylindrical shells, which can be effectively used in qualitative investigations on the instability phenomena of thin-walled structural members.(3) The formulation expressed in terms of stresses and strains is adopted instead of resultant forces, so that all kinds of constitutive equations can be easily introduced and the application to general material-nonlinear problems such as visco-elasto-plastic problems is possible.

Explicit and implicit methods for design sensitivity analysis of nonlinear structures subjected to transient dynamic loads are described. The direct differentiation method is used to calculate the sensitivities. Elastic and elasto-plastic problems undergoing large deformation are treated in the formulation. The explicit method is quite straightforward to implement for analysis as well as design sensitivity analysis: knowing all the quantities at time t, one updates the quantities for the next time point without solution of any linear system of equations, or any iterations. For the implicit method, the effective stiffness matrix for the sensitivity equation is different from that for the analysis phase due to inclusion of large deformation in the formulation. A numerical procedure to integrate the equation is proposed which uses the symmetric effective stiffness matrix from the analysis phase. Although the sensitivity analysis problem is linear, the procedure requires iterations for sensitivity calculations at each time step, as for the analysis phase. The implicit method is implemented using the pre-conditioned conjugate gradient iterative solution scheme. Numerical results for sensitivities indicate the implicit method to be more accurate than the explicit method for transient elastoplastic problems. However, the explicit method is substantially more efficient than the implicit method.

A finite element model to the analysis of thin shell structures with cutouts/inclusions is developed based on shear deformation theory. An eight-noded isoparametric shell element is considered with six degrees of freedom at each node. A convergence study is under taken to arrive at a suitable finite element mesh. The loading conditions include uniform tension, uniform pressure. The finite element results presented in this work are shown to be in good agreement with available analytical and experimental data and the direction for future work is also given.

An eighteen-node, three-dimensional, solid element is presented for the finite element analysis of thin shell structures. The element is based on the Hellinger-Reissner principle with assumed independent strain divided into higher and lower order terms. The stiffness matrix associated with the higher order independent strain plays the role of a stabilization matrix. A modified stress-strain relation decoupling inplane and normal strain is used to incorporate thin shell behavior. Numerical results demonstrate that with a properly chosen set of assumed strain this element is effectively free of locking even for very thin plates and shells.

Further development of three dimensional tri-linear elements based on a modified enhanced strain methodology is presented. The new formulation employs Taylor expansions of the derivatives of the isoparametric and enhanced shape functions in local coordinates. With this approach, only nine enhanced modes are needed for developing a dilatational-locking free element. Furthermore, the formulation permits a symbolic integration of the element tangent matrix, and more efficient static condensation procedure due to uncoupling of the enhanced modes. Good results in the analysis of thin shell structures, using only one 3D element in the thickness direction of the shell, are also presented.

In the unpredictable world of various natural phenomena under which the shell structures are designed to function, their intended strength and dynamic stability may become negotiable when subjected to unknown parameters. This dilemma for the structural engineer is a common feature of most marine shell constructions. The general governing equations of equilibrium and dynamic stability for thin shell structures subjected to follower loads and undergoing large deformations have already been developed for the general shell defined in a monoclinically convected coordinate system.In a previous report, the complex peninsular shaped instability regions of a shell at various disturbed equilibrium positions have been numerically illustrated. This paper expands on the same direction and reaches at a demonstration of the disturbance threshold curves for the dynamic stability of a shell. The numerical results for subsequent equilibrium positions show that the shell stability slips suddenly over a threshold point of dynamic equilibrium from the heteronomous state to an autonomous state of self-sustained motions. Beyond this point the dynamic behaviour becomes highly irregular until the static critical point is reached, at which deep shells generally fail and the shallow shells enter the next phase of post-collapse stability.

Direct differentiation method for response sensitivity analysis of a bounding surface plasticity soil model

Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

The objective of this study is to examine two analytical methods for the design sensitivity analysis of optimal control problems and to develop the analyses necessary for the application of the methods to thermal systems. The two methods are the direct differentiation method and the adjoint variable method. Results for these methods are compared with those obtained using central finite difference approximations for design sensitivity analysis. The adjoint variable method has the potential to reduce significantly computational times for design sensitivity analysis while sacrificing only a small amount of accuracy. As the number of functional requiring design sensitivity analysis increases, however, the adjoint variable method becomes less attractive and the direct differentiation method becomes more attractive.

Effect of buckling-restrained brace model parameters on seismic structural response

Using the direct differentiation method, a design sensitivity analysis method for time-dependent incompressible fluids is developed. The fluid behavior is described as the motion of particles involved by the SPH method. In the SPH projection method, instead of changing the fluid density, incompressibility is enforced by the pressure Poisson equation derived from pressure projection, which enable to use larger time steps. In spite of the additional pressure Poisson equation, the computational cost for the design sensitivity is not expensive since the factorized system matrix of pressure Poisson equation can be utilized. Aforementioned computational efficiency is very beneficial for the design sensitivity computation required for every time step in explicit time integration and updated Lagrangian schemes, for which an update scheme of design velocity field is developed using the velocity sensitivity. Through demonstrative numerical examples, the developed DSA method turns out to be efficient and shows excellent agreement with finite differencing.

Design sensitivity analysis of dynamic response of nonviscously damped systems