This chapter is devoted to modeling the properties of composite materials and structures. Mathematical relations describing the nonlinear elastic three-point bending of isotropic and reinforced beams with account of different strength and stiffness behavior in tension and compression are obtained. An algorithm for numerical solution of corresponding boundary-value problems is proposed and implemented. Results of numerical modeling were compared to acquired data for polymer matrix and structural carbon fiber reinforced plastics. A computational technology for analysis and optimization of composite pressure vessels was developed and presented.

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Mathematical Modeling and Numerical Optimization of Composite Structures

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Solving the biharmonic equation in irregular domains by the least squares collocation method

This paper is devoted to modeling the properties of composite materials and reinforced composite beams. Mathematical relations describing the nonlinear elastic three-point bending of isotropic and reinforced beams with account of different strength and stiffness behavior in tension and compression are obtained. An algorithm for numerical solution of corresponding boundary-value problems is proposed and implemented. Results of numerical analysis were compared to acquired data for polymer matrix and structural carbon fiber reinforced plastics.

Mathematical modeling and numerical analysis of reinforced composite beams

The equations of the spatial theory of elasticity of bending of layered plates are solved with the help of an asymptotic splitting method. This method was successfully used in problems of longitudinal-transverse bend of beams. Boundary-layer theory is used to describe material behavior near the beam edges.

Метод асимптотического расщепления пространственной теории слоистых пластин

Fluid–structure interaction with the spectral method: application to a cylindrical tube subjected to transverse flow

Least squares collocation method (LSC) is a versatile numerical method for solving boundary value problems for PDE. The present article demonstrates the abilities of LSC to solve various problems – in particular, calculations of bending of isotropic irregular shaped plates and multi-layered anisotropic plates. In order to achieve higher accuracy, new versions of the method utilize high-degree polynomial spaces. The numerical experiments demonstrate high accuracy of the solutions.

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New Possibilities and Applications of the Least Squares Collocation Method

h-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications

Vasiliy Shapeev

Papers

New Possibilities and Applications of the Least Squares Collocation Method

h-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications