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MIT2_094S11_2094_lectures

MIT2_094S11_2094_lectures - 2.094 Finite Element Analysis...

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2.094— Finite Element Analysis of Solids and Fluids Fall ‘08 MIT OpenCourseWare
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Contents 1 Large displacement analysis of solids/structures 3 1.1 Project Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Large Displacement analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Mathematical model/problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Requirements to be fulfilled by solution at time t . . . . . . . . . . . . . . . . . . . 5 1.2.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Finite element formulation of solids and structures 7 2.1 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Finite element formulation for solids and structures 10 4 Finite element formulation for solids and structures 14 5 F.E. displacement formulation, cont’d 19 6 Finite element formulation, example, convergence 23 6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.1.1 F.E. model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.1.2 Higher-order elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 Isoparametric elements 28 8 Convergence of displacement-based FEM 33 9 u / p formulation 37 10 F.E. large deformation/general nonlinear analysis 41 11 Deformation, strain and stress tensors 45 12 Total Lagrangian formulation 49 1
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MIT 2.094 Contents 13 Total Lagrangian formulation, cont’d 53 14 Total Lagrangian formulation, cont’d 57 15 Field problems 61 15.1 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 15.1.1 Differential formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 15.1.2 Principle of virtual temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 15.2 Inviscid, incompressible, irrotational flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 16 F.E. analysis of Navier-Stokes fluids 65 17 Incompressible fluid flow and heat transfer, cont’d 71 17.1 Abstract body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 17.2 Actual 2D problem (channel flow) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 17.3 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 17.4 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 17.5 FSI briefly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 18 Solution of F.E. equations 76 18.1 Slender structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 19 Slender structures 81 20 Beams, plates, and shells 85 21 Plates and shells 90 2
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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 1 - Large displacement analysis of solids/structures Prof. K.J. Bathe MIT OpenCourseWare 1.1 Project Example Physical problem Reading: Ch. 1 in the text “Simple” mathematical model analytical solution F.E. solution(s) More complex mathematical model holes included large disp./large strains F.E. solution(s) How many finite elements? We need a good error measure (especially for FSI) “Even more complex” mathematical model The“complex mathematical model”includes Fluid Structure Interaction (FSI). You will use ADINA in your projects (and homework) for structures and fluid flow. 3
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MIT 2.094 1. Large displacement analysis of solids/structures 1.2 Large Displacement analysis Lagrangian formulations: Total Lagrangian formulation Updated Lagrangian formulation Reading: Ch. 6 1.2.1 Mathematical model/problem Given the original configuration of the body, the support conditions, the applied external loads, the assumed stress-strain law Calculate the deformations, strains, stresses of the body. Question Is there a unique solution? Yes, for infinitesimal small displacement/strain. Not necessarily for large displacement/strain.
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