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In this paper, we present a simultaneous optimization method of shape and topology for designing a light-weight plate and shell structure. The free-form optimization method for shells and SIMP method are respectively employed for shape and topology optimization, and combined effectively. Shape and fictitious homogenized-density variations are used as the design variables, and simultaneously determined in one iteration of the convergence process. With this method, the optimal topology is determined in the variable design surface optimized by shape optimization. Compliance is used as the objective functional, and minimized under the volume constraint. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined by the H1 gradient method, where the sensitivity functions are applied as the Robin condition to the design surface. With the proposed method, the optimal lighter and stiffer shell structure with smooth surface can be obtained without any design parameterization and numerical instabilities such as checkerboard and zigzag-shape problems.

In this study, we propose an optimization method for the reaction forces control problem of shell structures in plastic range. The sum of squared error norms between reaction forces in the plastic range and the target forces are minimized under a volume constraint. The total strain theory is employed for simplicity under the assumption that the considered force or enforced displacement increases monotonically, so that the final deformation is path-independent. The optimum shape design problem is formulated as a distributed-parameter system, and we assume that a shell is varied in the out-of-plane direction to the surface whereas the thickness is not varied during the shape change. The shape gradient function and the optimality conditions for this problem are derived theoretically using the material derivative method and the adjoint variable method. The shape gradient function calculated from a non-linear finite-element analysis is applied to the H 1 gradient method for shells proposed by one of the authors to determine optimum shape variation. The optimal shape of a shell structure can be obtained without shape parameterization while maintaining surface smoothness. Numerical examples are presented to demonstrate the validity and practical utility of the proposed free-form optimization method. [ABSTRACT FROM AUTHOR]

In the present work, in order to tackle the linear elastic buckling problem, we develop a shape optimization process based on a free-form optimization method to optimize a shell structure under out-of-plane and in-plane shape variations. The free-form optimization method is a node-based method in which mesh regularity can be maintained and shape design parameterization is not required. It has the advantages of sufficient efficiency for treating large-scale problems and the ability to realize a smooth shape. The buckling coefficient in buckling mode 1 is set as the objective function in the shape optimization problem. We consider repeated eigenvalues and volume constraint for the whole optimization process. Three numerical examples are presented in this work to illustrate the shape optimization algorithm and to show that it can increase the buckling strength substantially, especially under shape variation in the out-of-plane direction. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-phase optimization method for designing the shape and thickness of shell structures consisting of orthotropic materials. We consider a multi-objective in terms of the compliances under multi boundary conditions, and use the weighted sum compliance as the objective functional and minimize it under the volume and the state equation constraints. The 1st phase is shape optimization, in which a shell structure is varied in the out-of-plane direction to the surface to create its optimal shape. In the 2nd phase, thickness optimization is implemented after shape optimization to decrease the compliance further. A free-form shape and thickness optimization problem is formulated in a distributed-parameter system based on the variational method. The shape and thickness sensitivities are theoretically derived and applied to the H 1 gradient method for shape and size optimization. The optimal multi-objective free-form of a shell structure with an orthotropic material can be determined using the proposal method, and the influence of the orthotropic angle to the optimal shape and thickness distribution is investigated in detail. [ABSTRACT FROM AUTHOR]

This paper presents a free-form optimization method for maximizing the elastic buckling load of a shell structure. A shell structure is assumed to be movable in the normal or the tangent direction to the surface for the shape optimization. The 1st buckling load factor is maximized under a volume constraint. The 1st eigenvalue constraint is employed to avoid the repeated eigenvalues problem in this max-min problem. This problem is formulated as a distributed-parameter shape optimization problem, and the shape gradient function for this problem is theoretically derived using the Lagrange multiplier method, the adjoint variable method and the formulae of the material derivative. The shape gradient function derived is applied to the free-form optimization method for shells, a gradient method in the Hilbert space, where the optimal shape variation is calculated as the displacement field of the linear elastic analysis of the fictitious shell model. With this method, the optimal free-form surface or boundary is determined without shape parameterization. The effectiveness of the 1st eigenvalue constraint is also studied. The results show the validity of this method to determine the optimal free-form of shell structures for the elastic buckling design problem.

This paper describes a parameter-free free-form optimization method for shell structures to reduce the noise in a closed acoustic field. Squared sound pressure, which is evaluated by solving fully coupled shell-acoustic interaction system, is selected as the objective function to be minimized. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that shell structures are varied in the out-of-plane direction to the surface and the thickness is constant. The shape gradient function and the optimality conditions are derived by using the material derivative method, and they are applied to the free-form optimization method for shells. The shape gradient function is applied to the shell surface as a fictitious distributed force under Robin condition for varying the surface, for minimizing the objective functional and for regularizing the mesh. With this method, the smooth optimal free-form of shell structures are obtained without any shape design parameterization. The validity of this method for the radiated noise reduction from vibrating shell structures in a closed space is verified through design examples.

This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for the optimum free-form design of shell structures. [ABSTRACT FROM AUTHOR]

In a free-form shell structure, the possibilities of masonry construction beyond vertical plane wall elements are explored. The shape has been developed in an iterative process, employing sculptural design methods and numerical modelling to obtain the final stable shell shape. In the construction in full scale, the practical feasibility could be demonstrated. The project shows the capability of masonry to assume complex shapes, and how this can be achieved using current building material and fulfilling the codes. In particular, the feasibility of shell structures with pronounced double curvature is shown. A further aspect of the project is the design process, with the attempt to establish a continuous information flow from design to realization, by means of Information Technology - integrating shape development on physical models and CAD, numerical modelling, and realization. Design and realization took place as seminars with students of Architecture, following the didactic approach of "learning by doing". [ABSTRACT FROM AUTHOR]