Keywords: finite element, neural network, linear solver, stiffness matrix, element stiffness simulation, SCNN.

The Finite Element Method (FEM) is an important method in numerical computations. It has been successfully applied in fields of solid, fluid and electromagnetism. To simulate FEM analysis with artificial neural network is a challenge problem to both mechanics and computer science. Since it is a new way to solve some kinds of engineering problems, such as structural analysis with unknown or complicated constitutive relation components, linear analysis for structure with non-linear components, real time analysis and fast re-analysis. The ESS approach was firstly presented by the author[1] to solve these problems. This approach simulates the relationship between displacements and loads of an element using an artificial neural network (ANN). A hybrid stiffness matrix is constructed with the ANN and other conventional element stiffness matrix. The hybrid matrix can be constrained, trained and solved[1]. So we can get a global solution with some unknown elements. This method is successful but still have some difficulties. One is to find a stable solver to the ESS approach. In this approach, the global stiffness matrix contains some neural network elements, and is usually not positive and symmetry. Therefore, the conjunctive gradient method, which we applied before, is not always applicable.

Structure controllable multi-layer forward neural network (SCNN) has been applied to solve matrix inversion[5,3,4], linear equations[2,3,4], eigenvalues/eigenvectors[6] and related problems. It shows the ability to solve illness condition equations. In this paper, we construct a SCNN solver to solve the hybrid stiffness matrix, and it works in all cases. Furthermore, not only the static problems but also the dynamic problems can be solved by the SCNN. Learning factor a is a key parameter to the efficiency of improved BP algorithm. Deduction of this parameter is also given in this paper.

By using the SCNN, accuracy of present results increase dramatically comparing to the previous. Additionally, eigenvalues and eigenvectors of structure can be solved by SCNN in ESS approach. The simulation results are satisfactory. So we believe the SCNN is a significant solver for finite element analysis with neural network elements.

1

Li S., "Global flexibility simulation and element stiffness simulation in finite element analysis with neural networks", Comput. Methods Appl. Mech. Engrg, 186(1), 2000. doi:10.1016/S0045-7825(99)00108-5

2

Cichocki A. and Unbehauen R., "Neural network for solving systems of linear equations and related problems", IEEE Trans. CAS, 39, 124-138, 1992. doi:10.1109/81.167018

3

Wang L. and Mendel J.M., "Matrix computations and equation solving using structured networks and training", Proc. the 29th CDC, Hauill, Dec 1990. doi:10.1109/CDC.1990.203920

4

Wang L. and Mendel J.M., "Structured trainable networks for matrix algebra", Proc. IJCNN 92, Washington, 1992. doi:10.1109/IJCNN.1990.137705

5

Luo F. and Bao Z., "Neural network approach to computing matrix inversion", Applied Math. and Computation, 47, 109-120, 1992. doi:10.1016/0096-3003(92)90040-8

6

Samadzija N. and Waterland R., "A neural network for computing eigenvectors and eigenvalues", Biol. Cybem., 65, 211-214, 1991. doi:10.1007/BF00206218

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