A MATLAB-Based Symbolic Approach for the Quick Developing of Nonlinear Solid Mechanics Finite Elements

A symbolic mathematical approach for the rapid early phase developing of finite elements is proposed. The algebraic manipulator adopted is MATLAB® and the applicative context is the analysis of hyperelastic solids or structures under the hypothesis of finite deformation kinematics. The work has been finalized through the production, in an object-oriented programming style, of three MATLAB® classes implementing a truss element, a tetrahedral element and plane element. The approach proposed, starting from the mathematical formulation and finishing with the code implementation, is described and its effectiveness, in terms of minimization of the gap between the theoretical formulation and its actual implementation, is highlighted.

Wave Motion Analysis in Plane via Hermitian Cubic Spline Wavelet Finite Element Method

A plane Hermitian wavelet finite element method is presented in this paper. Wave motion can be used to analyze plane structures with small defects such as cracks and obtain results. By using the tensor product of modified Hermitian wavelet shape functions, the plane Hermitian wavelet shape functions are constructed. Scale functions of Hermitian wavelet shape functions can replace the polynomial shape functions to construct new wavelet plane elements. As the scale of the shape functions increases, the precision of the new wavelet plane element will be improved. The new Hermitian wavelet finite element method which can be used to simulate wave motion analysis can reveal the law of the wave motion in plane. By using the results of transmitted and reflected wave motion, the cracks can be easily identified in plane. The results show that the new Hermitian plane wavelet finite element method can use the fewer elements to simulate the plane structure effectively and accurately and detect the cracks in plane.

Higher-order assumed strain plane element immune to mesh distortion

Purpose This paper aims to propose a new robust membrane finite element for the analysis of plane problems. The suggested element has triangular geometry. Four nodes and 11 degrees of freedom (DOF) are considered for the element. Each of the three vertex nodes has three DOF, two displacements and one drilling. The fourth node that is located inside the element has only two translational DOF. Design/methodology/approach The suggested formulation is based on the assumed strain method and satisfies both compatibility and equilibrium conditions within each element. This establishment results in higher insensitivity to the mesh distortion. Enforcement of the equilibrium condition to the assumed strain field leads to considerably high accuracy of the developed formulation. Findings To show the merits of the suggested plane element, its different properties, including insensitivity to mesh distortion, particularly under transverse shear forces, immunities to the various locking phenomena and convergence of the element are studied. The obtained results demonstrate the superiority of the suggested element compared with many of the available robust membrane elements. Originality/value According to the attained results, the proposed element performs better than the well-known displacement-based elements such as linear strain triangular element, Q4 and Q8 and even is comparable with robust modified membrane elements.

Using Higher-Order Strain Interpolation Function to Improve the Accuracy of Structural Responses

It is widely known that the accuracy of the finite element method has a direct relation with the type of elements and meshes. Another issue which has remained less treated is the impact of loading type on the accuracy of responses. Changing the applied forces from concentrated to distributed loading has a great effect on the accuracy of certain types of elements and this action can greatly reduce their accuracy. Particularly in the coarse meshes, it creates a critical situation. Some elements do not have the ability to provide the exact answers in stated conditions. For example, the well-known plane element, LST, demonstrates promising performance under concentrated shear and bending loading as well as surface traction. In the case of distributed loads and coarse meshes, its accuracy diminishes considerably. To remedy this defect, in this paper, a new higher-order triangular element is proposed by using natural assumed strain approximation. Various numerical examples demonstrate high accuracy and efficiency of the element in comparison with common well-known finite elements in analysis of structures under distributed loading.

Closed-Form Evaluation of Potential Integrals in the Boundary Element Method

A method is presented for the closed-form evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such integrals on a plane element, and used to develop a criterion for the selection of quadrature rules. The analytical approach is based on an optimized expansion of the Green’s function for the problem, selected to limit the error to some required tolerance. Results are presented showing accuracy to tolerances comparable to machine precision.