Keywords: shells, hexahedral, locking, benchmarks, optimisation, sensitivities, design elements concept.

This paper represents the first stage of a project in which shell structures are proposed as the basis given that they offer the greatest potential to explore the relationship between form and structural mechanics, and from a practical engineering perspective intrinsically offer the most obvious opportunity for design ingenuity and individualism. Furthermore, the "form" is defined by both thickness of the shell and abscissa geometry. Morphology is defined here to refer to both section (e.g. thickness) and global (e.g. form) geometries. Within the optimization research community researchers work either on topology optimization or shape optimization. In this research we aim to combine both topology and shape optimization into an integrated shell "morphology".

Shell formulation used in this paper is a linear 18-node hexahedral shell element with assumed natural strain locking alleviation strategies to eliminate membrane and transverse shear locking. Thickness and trapezoidal locking are eliminated using a modified constitutive matrix and assumed natural transverse normal strains. The resulting new element formulation passes several widely used benchmarks for example a hemisphere shell with an 18 opening subjected to two equal and opposite concentrated loads as depicted in figure 1a. Normalized displacements at the load points are shown in figure 1b and compared with results from semiloof and 9-node Lagrangian [1] and shown to be rapidly convergent to within 1% of the exact solution.

A significant challenge in a shape optimization problem is that the geometry of the structure may change substantially in every iteration during the optimization process. This means that using the same finite element mesh may cause distortion which may lead to inaccurate results from finite element analysis. Therefore re-meshing is required after each iteration of the optimization process. To save computational time, automatic mesh generation schemes using the design elements concept and isoparametric technique have been proposed in conjunction with degenerated shell elements. In this method, structures are divided into design elements [2]. These design elements consist of master nodes that define shapes of the elements. Each design element is then divided into finite element mesh preserving geometric regularity and avoiding distortion. Coordinates of finite element nodes in design elements can be defined as,

in which , and are shape functions and number of master nodes, respectively. Analytical design sensitivities are also derived assuming the design elements concept at the equilibrium configuration as in

where and are stiffness matrices, displacements, design variables and load vectors, respectively. Design variables, , are assumed to be the master nodes and the shell thickness. The sensitivity requires , as,

where and are strain-displacement, constitutive matrix and determinant of Jacobian matrix, respectively. The novel application of the design elements concept, in which a temporary mid-surface is established and the upper and lower surfaces of the shell defined by a corresponding normal, as presented in this paper, enables the coupled "morphology" sensitivities to be derived. Coupling is achieved through the terms and as presented in the paper. Validity of the sensitivities are demonstrated using finite difference approximations.

1

White DW., Abel JF. Testing of shell finite element accuracy and robustness. Fin. Elem. Anal. Des. 1989;6: 129-51 doi:10.1016/0168-874X(89)90040-1

2

Imam MH. Three-dimensional shape optimiz. Int. J. Num. Meth. Eng. 1982;18:661-73 doi:10.1002/nme.1620180504

3

Huang HC., Hinton E. A new nine node degenerated shell element with enhanced membrane and shear interpolation. Int. J. Num. Meth. Eng. 1986;22:73-92 doi:10.1002/nme.1620220107

4

Hauptmann R., Schweizerhof K. A systematic development of 'solid-shell' element formulations for linear and non-linear analyses employing only displacement degrees of freedom. Int. J. Num. Meth. Eng. 1998;42: 49-69 doi:10.1002/(SICI)1097-0207(19980515)42:1<49::AID-NME349>3.3.CO;2-U

5

Bischoff M., Ramm E. Shear deformable shell elements for large strains and rotations. Int. J. Num. Meth. Eng. 1997;40: 4427-49 doi:10.1002/(SICI)1097-0207(19971215)40:23<4427::AID-NME268>3.0.CO;2-9

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