Mètodes Numèrics: A First Course on Finite Elements–In the p-version of the finite element...
Transcript of Mètodes Numèrics: A First Course on Finite Elements–In the p-version of the finite element...
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Mètodes Numèrics: A First Course on Finite Elements
Dept. Matemàtiques
ETSEIB - UPC BarcelonaTech
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Short History: annals
• 1943: Richard Courant, a mathematician, described a
piecewise polynomial solution for the torsion problem of ashaft of arbitrary cross section. Even with holes.The early ideas of FEM date back to a 1922 book by Hurwitz
and Courant.
1888-1972: born in Lublintz GermanyStudent of Hilbert and Minkowski in Gottingen GermanyPh.D in 1910 under Hilbert’s supervision.1934: moved to New York University, founded the Courant Institute
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Short History: name
• 1960: The name "finite element" was coined by structural
engineer Ray Clough of the University of California.
1920: born in SeattleProfessor emeritus of Structural Engineering at UC BerkleyStudied for Boeing the vibration of wing airplanes Ph.D from MIT. Well known as a earthquake engineer
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Short History: divulgation
• 1967: Olgierd C. Zienkiewicz published The first book on
the FEM O.C. ZIENKIEWICZ (with Y.K. CHEUNG), (1967) The Finite Element Method in Continuum and Structural Mechanics , McGraw Hill, 272 pp (translated into Japanese and Russian)
1921-2009: born in Caterham, England. Professor Department of Civil Engineering at Swansea UniversityPh.D from Imperial College London. He published nearly 600 papers and wrote or edited more than 25 books
For more information:https://en.wikipedia.org/wiki/Finite_element_method
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Short History: FEM-Mathematics
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The Finite Elements Approach
Dept. Matemàtiques
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Following:
http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/
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General Idea:
Divide and Conquer!!
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General Idea: Divide and Conquer
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General Idea: Divide and Conquer
• Subdivide the domain in different parts:
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General Idea: Divide and Conquer
• This can be done in several different fields:
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General Idea: Example
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General Idea: Example
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Generalized Approach
• Step 1: State the physical problem
STRUCTURAL
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Generalized Approach
• Step 1: State the physical problem
STRUCTURAL
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Generalized Approach
• Step 1: State the physical problem
THERMAL
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Generalized Approach
• Step 1: State the physical problem
FLUIDS
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Generalized Approach
• Step 1: State the physical problem
AERODYNAMICS
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Generalized Approach
• Step 1: State the physical problem
Electromagnetism
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Generalized Approach
• Step 1: State the physical problem
• Step 2: State the Mathematical model
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Generalized Approach
• Step 2: State the Mathematical model
Partial Differential Equations (PDE)
Heat Transfer (1-D):𝜕𝑢
𝜕𝑡= 𝑘
𝜕2𝑢
𝜕𝑥2𝑢 = 𝑢(𝑡, 𝑥) temperature𝑘 thermal conductivity coeff.
𝑢𝑡 = 𝑘𝑢𝑥𝑥
Analogous notation:
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Generalized Approach
• Step 2: State the Mathematical model
Partial Differential Equations (PDE)
Wave equation (1-D):𝜕𝑢
𝜕𝑡2= 𝑐2
𝜕2𝑢
𝜕𝑥2𝑢 = 𝑢(𝑡, 𝑥) high value𝑐 velocity wave propagation
𝑢𝑡𝑡 = 𝑐2𝑢𝑥𝑥
Analogous notation:
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Generalized Approach
• Step 2: State the Mathematical model
Partial Differential Equations (PDE)
Fluid equations (2-D): Navier-Stokes
(𝑢𝑥, 𝑢𝑦) fluid velocity components
𝜇 viscosity 𝜌 density𝑝 pressure
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Generalized Approach
• Step 2: State the Mathematical model
Partial Differential Equations (PDE)
Structural Analysis (2-D): Plane Stress
(𝑢, 𝑣) displacement components𝜈 poisson ratio
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Generalized Approach
• Step 1: State the physical problem
• Step 2: State the Mathematical model
• Step 3: Numerical Solution
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Generalized Approach• Step 3: Numerical Solution
3.1 Domain discretization: Choose elements
Nodes
Elements
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Generalized Approach• Step 3.1 Domain discretization:
One-Dimensional
Frame Elements
Two-Dimensional
Triangular Elements
Three-Dimensional
Brick Elements
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Generalized Approach• Step 3.2. Numerical solution (Linear Systems)
– For each element we obtain a small linear system
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Generalized Approach• Step 3.2. Numerical solution (Linear Systems)
– For each element we obtain a small linear system
Exemple: Structural 1-dim bar:
𝑓𝑒 = 𝐾𝑒𝑢𝑒
𝐾𝑒 =𝐸𝐴
𝐿1 −1−1 1
𝑢𝑒: displacements, 2 𝑓𝑒: forces, 2 𝐾𝑒: Stress Matrix, 2x2
E: Young Modulus (elasticity coef.)A: section Area of the barL: Length of the bar
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Generalized Approach• Step 3.2. Numerical solution (Linear Systems)
– Assembling all elements we obtain a BIG global linear system
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Generalized Approach
• Step 1: State the physical problem
• Step 2: State the Mathematical model
• Step 3: Numerical Solution
• Step 4: Solution Postprocess
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Generalized Approach• Step 4: Solution Postprocess
– Compute other magnitudes (Stress, thermal flow, etc)
– Plot results
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Generalized ApproachSummary
Physical Problem (Modeling Equations)
Domain discretization(Elements, Nodes)
Problem Solution (Linear Systems)
Solution Postprocess(Show new results)
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Generalized ApproachSummary
Physical Problem (Modeling Equations)
Domain discretization(Elements, Nodes)
Problem Solution (Linear Systems)
Solution Postprocess(Show new results)
Mathematical tools
Partial Differential Equations
Weak Problem Formulation
Shape functions (interpolation)
Numerical Integration
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Generalized ApproachSummary
Physical Problem (Modeling Equations)
Domain discretization(Elements, Nodes)
Problem Solution (Linear Systems)
Solution Postprocess(Show new results)
Mathematical steps
Element dimension and shape
Meshing general domains
Impose Boundary Conditions
Obtain element matrices
Enumerate Element and Nodes
Global Assembly
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Generalized ApproachSummary
Physical Problem (Modeling Equations)
Domain discretization(Elements, Nodes)
Problem Solution (Linear Systems)
Solution Postprocess(Show new results)
Mathematical steps
Matrix shape and reordering
Vector and Matrix Manipulation
Numerical methods for Linear Systems
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Generalized ApproachSummary
Physical Problem (Modeling Equations)
Domain discretization(Elements, Nodes)
Problem Solution (Linear Systems)
Solution Postprocess(Show new results)
Mathematical steps
Compute secondary variables
Plot results
Evaluate solution: critical points, remeshing, estimate errors, etc.
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Error Sources• Simplified Geometry
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Error Sources• Simplified Geometry
• Shape function approximation (polynomial)
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Error Sources• Simplified Geometry
• Shape function approximation (polynomial)
• Numerical integration (Gauss Quadrature)
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Solution convergence strategies
• How to minimize the error?
– Actual solution is not known
– Strategy:
• Compare between two different results of the same problem
• Check how much it differs
• Take a third result. There is convergence?
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Solution convergence strategies• H-method
Consist in reducing the size of the elements in order to obtain a better approximation of the domain.
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Solution convergence strategies• P-method
Consist in increasing the order of the interpolation shape functions (use more nodes) in order to obtain a better approximation of the deformation solution.
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Solution convergence strategies• Which method is better? There’s no a unique conclusion
– In the p-version of the finite element method the rate ofconvergence cannot be slower than in the h-version.
– Numerical oscillation problem would happen for p-version of thefinite element method.
– For obvious practical reasons, finite element analyses should beboth efficient and reliable.
• As a general point of view:– For structural analysis, h-method is more popular. Two-order
element is a good application considering the efficiency andreliability.
– p-method seems to act against the original goal of FEM