Keywords: spherical geometry, MEMS, elastostatics, plasticity, spherical harmonics.

The modeling of axispherical structures is normally carried out using axicylindrical finite element meshes. When unsymmetrical loads are encountered, the modeling of spherical bodies becomes far from trivial. It can be accomplished using axicylindrical formulations that provide for unsymmetrical loading conditions via a Fourier series description of the load. (Hence, even though the geometry is axicylindrical, all of the displacements of the 3-D continuum are provided for.) However, for a sphere, axicylindrical meshes are inevitably prone to element distortion, while the distance to the axis of symmetry varies dramatically for different elements in the mesh. As a result, convergence rates are low.

Spherically symmetric loads applied to an isotropic sphere result in spherically symmetric displacement fields within the sphere. Since only one non-zero displacement interpolation function, namely , is required to model this condition, a one-dimensional element can be used.

In our paper, we present efficient axispherical finite element formulations for the numerical modeling of spherically symmetric structures. The formulation results in an efficiently banded global stiffness matrix . The solution of the equilibrium equations is therefore far less computationally expensive than for axicylindrical or solid models. Formulations are presented, which provide for elastostatic deformation [1], piezoelectric effects [2,3] and material non-linearity in the form of elastic-perfect-plastic behavior [4]. Provision is made for unsymmetrical loading conditions via a suitable series description of the applied loads using spherical harmonics [5].

For the sake of brevity, we focus on the coupling between mechanical, electrostatic and piezoelectric effects during numerical evaluation of the formulations. The advantages and ease of formulation of the proposed formulations as compared to axicylindrical models are emphasized. Said advantages include:

• Dramatic reductions in computational effort when solving the equilibrium equations are obtained.
• No distortion effects in the element.
• Highly increased accuracy when modeling spherically symmetric structures.
• Ease of formulation, implementation and modification.
• Ease of inclusion of multiphysics phenomena.

One possible application of our novel element is the modeling of spherical micro-electro-mechanical systems (MEMS). In recent times, MEMS have received significant attention. They are lightweight, ultra small devices, which are widely used for integrated sensor applications, e.g. acceleration sensors for airbags, active actuators for micro air vehicles (MAV's), etc. They have potential applications in micro-surgery, health-monitoring, micro-fabrication, etc.

Spherical MEMS devices are perceived to be important in a number of applications. They are unidirectional, an attractive feature when the to-be-sensed quantity is of unknown location in an effectively infinite field. Spherical MEMS are, for example, used on a very large scale in active and passive sonar arrays.

While many an axispherical problem can of course be solved analytically, this becomes very difficult, and even infeasible, when multiphysics phenomena and, for example, plasticity effects are considered. The accommodation of asymmetric loads via spherical harmonics further increase the applicability of the one-dimensional elements.

1

K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, 1982.

2

S.M. Peelamedu, A.R. Barnett, R.V. Dukkipati, and N.G. Naganathan. Finite element approach to model and analyze piezoelectric actuators.
J. Intelli. Mater. Syst. Struct., 44:476-485, June 2001.

3

H. Allik and T. J. R. Hughes. Finite element method for piezoelectric vibration.
Int. J. Num. Meth. Eng., 2:151-157, 1970. doi:10.1002/nme.1620020202

4

J. Lubliner. Plasticity Theory. Macmillan Publishing Company, 1990.

5

D. Jackson. Fourier Series and Orthogonal Polynomials. Carus Mathematical Monographs, No. 6. Mathematical Association of America, Menasha, Wisconsin, 1941.

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