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Steak, shake or break – and other applications of FEM TMA4220

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Outline Present the governing equations (3 of them) Present the governing equations (3 of them) Building your FEM code Building your FEM code Mesh generation Mesh generation Experimentation Experimentation Integration with other courses Integration with other courses

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Time dependent Poisson “Steak”: Heat transfer “Steak”: Heat transfer The way heat propagates through objects The way heat propagates through objects

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Linear elasticity “Break”: Stress analysis “Break”: Stress analysis How things bend and deform How things bend and deform

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Vibration analysis “Shake”: Eigenmode analysis “Shake”: Eigenmode analysis Natural frequency of which things oscillate Natural frequency of which things oscillate

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Building your FEM code Series of problems 1a) – 3b) Gauss quadrature FEM theory Sample 2d problem Sample 3d problem Your choice and experimentation (Problem 4)

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How to work On problem 1-3: answer short and answer right On problem 1-3: answer short and answer right On problem 4: Elaborate as much as possible On problem 4: Elaborate as much as possible

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Mesh generation Lots of issues (probably the most understated part of FEM training) Lots of issues (probably the most understated part of FEM training) 3 options 3 options GMSH (free, easy-to-use mesh tool, readable output) GMSH (free, easy-to-use mesh tool, readable output) Pre-made by myself Pre-made by myself I’ll get back to #3 later I’ll get back to #3 later

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Experimentation Find your own problem Find your own problem Choose an equation Choose an equation Choose a geometry Choose a geometry Choose an objective Choose an objective Analyze the results Analyze the results Experiment with geometry, material properties and/or boundary conditions Experiment with geometry, material properties and/or boundary conditions

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Example 1: beef How to cook the optimal beef How to cook the optimal beef Choose an equation Choose an equation Choose a geometry Choose a geometry Choose an objective Choose an objective Optimality criterium? Optimality criterium? Too short cooking – raw and unedible Too short cooking – raw and unedible Too long cooking – dry and tasteless Too long cooking – dry and tasteless Too varying temperature – all of the above Too varying temperature – all of the above Experimentation Experimentation

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Example 2: Structural analysis Linear elasticity Linear elasticity Choose an equation Choose an equation Choose a geometry Choose a geometry Choose an objective Choose an objective Max stress lower than material critical point Max stress lower than material critical point Will it withstand gravity? Will it withstand gravity? Will it withstand external forces? (creepers?) Will it withstand external forces? (creepers?)

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Example 3: free vibration What are the natural frequencies of your geometry (and material)? What are the natural frequencies of your geometry (and material)? Choose an equation Choose an equation Choose a geometry Choose a geometry Choose an objective Choose an objective Find the natural frequencies Find the natural frequencies Resonance with enviromental frequencies (wind, noise etc) Resonance with enviromental frequencies (wind, noise etc) Sound/music (harmonic overtones) Sound/music (harmonic overtones) Purely visual Purely visual

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Integration with other courses Numerical linear algebra (TMA4205) Numerical linear algebra (TMA4205) Mathematical Modeling (TMA4195) Mathematical Modeling (TMA4195) Experts in Team (TMA4850) Experts in Team (TMA4850)

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Practicalities 2 persons to each group 2 persons to each group Deadline: 18 th Nov. Deadline: 18 th Nov. Written report and presentation Written report and presentation No min/max pages No min/max pages Access to computer lab Access to computer lab I’m away next week, Anne away the next 2 weeks I’m away next week, Anne away the next 2 weeks

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Summary Choose your own problem Choose your own problem Be creative Be creative Play on your strengths Play on your strengths Geometry modeling Geometry modeling Visualization Visualization Physical Experiments Physical Experiments CPU speed CPU speed

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