By G. R. Liu
This textbook offers an available and self-contained description of the Galerkin finite aspect technique for the 2 vital versions of continuum mechanics, temporary warmth conduction and elastodynamics, from formula of the governing equations to implementation in Matlab. The assurance follows an intuitive process: the salient positive aspects of every preliminary boundary price challenge are reviewed, together with an intensive description of the boundary stipulations; the strategy of weighted residuals is utilized to derive the discrete equations; and transparent examples are brought to demonstrate the tactic advent -- Mesh loose tools for engineering difficulties -- Mechanics of solids and buildings -- rules for vulnerable types -- MFree form functionality building -- point loose Galerkin procedure -- Meshless neighborhood Petrov-Galerkin technique -- aspect interpolation tools -- Mesh loose tools for fluid dynamics difficulties -- Mesh unfastened tools for beams -- Mesh loose equipment for plates -- Mesh unfastened equipment for shells -- Boundary mesh loose equipment -- Mesh loose tools coupled with different tools -- Implementation concerns -- MFree2D
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Extra info for Mesh-free methods: moving beyond the finite element methods
Theoretically, these equations can be applied to all other types of structures, such as trusses, beams, plates, and shells, because they are all made of 3D solids. However, treating all the structural components as 3D solids makes computation very expensive, and practically impossible. Therefore, theories for taking geometric advantage of different types of solids and structural components have been developed. Application of these theories in a proper manner can reduce analytical and computational effort drastically.
Mechanics for solids and structures is by itself a huge topic that needs more than a book to cover in detail. To provide a fairly clear and systematic discussion of this topic, we start with the formulation of 3D solids. Formulations for other types of structural components, beams, plates, shells, are then deduced from these general equations of 3D solids. 2. The surface of the solid is further divided into two types of surfaces. 1 Types of structural components. on which the displacements are prescribed is denoted Γu.
42) 1238_Frame_C03 Page 40 Wednesday, June 12, 2002 4:46 PM 40 Mesh Free Methods: Moving beyond the Finite Element Method where θ is the rotation in the x–y plane. The rotation can be obtained from the deflection of the neutral axis of the beam, v, in the y direction. 10 shows a small representative cell of length dx of the beam. The beam cell is subjected to external force by , moment Mz, shear force V, and inertial force ρA v˙˙ , where ρ is the density of the materials and A is the area of the cross section.