Extreme Learning Machine-Assisted Solution of Biharmonic Equations via Its Coupled Schemes

Obtaining the solutions of partial differential equations based on various machine learning methods has drawn more and more attention in the fields of scientific computation and engineering applications. In this work, we first propose a coupled Extreme Learning Machine (called CELM) method incorporated with the physical laws to solve a class of fourth-order biharmonic equations by reformulating it into two well-posed Poisson problems. In addition, some activation functions including tangent, gauss, sine, and trigonometric (sin+cos) functions are introduced to assess our CELM method. Notably, the sine and trigonometric functions demonstrate a remarkable ability to effectively minimize the approximation error of the CELM model. In the end, several numerical experiments are performed to study the initializing approaches for both the weights and biases of the hidden units in our CELM model and explore the required number of hidden units. Numerical results show the proposed CELM algorithm is high-precision and efficient to address the biharmonic equation in both regular and irregular domains

A lumped stress method for plane elastic problemsand the discrete-continuum approximation

This paper proposes a rational method to approximate a plane elastic body through a latticed structure composed of truss elements. The method is based on the introduction of a relaxed stress energy that allows an extension of the original problem to a larger space of admissible stress fields, including stresses concentrated along lines. Use is made of polyhedral approximations of the Airy stress function. The truss analogy is employed to obtain a displacement formulation. The paper includes several numerical applications of the method to sample problems, a numerical convergence study and comparisons with exact solutions and standard finite element approximations

An unconstrained mixed method for the biharmonic problem

In this work we present a finite element method for the biharmonicproblem based on the primal mixed formulation of Ciarlet and Raviart [A mixed finite element method for the biharmonic equation, in Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 125\u2013143]. We introduce a dual mesh and a suitable approximation of the constraint that enables us to eliminate the auxiliary variable with no computational effort. Thus, the discrete problem turns to be governed by a system of linear equations with symmetric and positive definite coefficients and can be solved by classical algorithms. The construction of the stiffness matrix is obtained by using Courant triangles and can be done with great efficiency