computational algorithms for thermoelastic...
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Computational algorithms for thermoelastic problem
Petr N. Vabishchevich, Maria V. Vasilyeva
SCTEMMJuly 8-10, 2013, Yakutsk
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Elasticity
The isotropic linear elasticity problem
µ∆u + (λ+ µ) grad div u = 0, x ∈ Ω,
u(x , t) = 0, x ∈ ΓD ,
(σ · n)(x , t) = f (x , t), x ∈ ΓN ,
(σ · n)(x , t) = 0, x ∈ ∂Ω \ ΓD \ ΓN .
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Temperature
Temperature problem
cρ∂T∂t− div(λ grad T ) = 0, x ∈ Ω,
∂T∂n
= 0, x ∈ ΓN ,
T = g(x , t), x ∈ ΓD ,
− λ∂T∂n
= α(T − Tout), x ∈ ΓR .
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Thermoelastic problem
Mathematical model
µ∇2u + (λ+ µ) grad div u − (3λ+ 2µ)αT grad(T − T0) = 0,
cε∂T∂t
+ (3λ+ 2µ)αTT0∂ div u∂t
= div (k grad T ) .
Boundary conditions:
σ(x) = g , x ∈ ΓuN , u(x) = f , x ∈ Γu
D ,
−k∂T∂n
(x , t) = a, x ∈ ΓTN , T (x , t) = 0, x ∈ ΓT
ND
Initial condition:T (x , 0) = T0, x ∈ Ω,
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Approximation by time
µ∇2u + (λ+ µ) grad div u − α grad T = 0,
c∂T∂t
+ α∂ div u∂t
= div(k grad T
),
where α = (3λ+ 2µ)αT , c = cε/T0, k = /T0.
Discrete-time form
µ∇2un+1 + (λ+ µ) grad div un+1 − α grad T n+1 = 0,
α div un+1 + cT n+1 − τ div(k grad T n+1
)= cT n + α div un.
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Weak formulation
Find T ∈ H(Ω),u ∈ H(div,Ω) such that
∫Ωσ(un+1) ε(v)dx −
∫Ωα(grad T n+1, v)dx =
∫ΓuN
(g , v)ds,∫Ωα div un+1qdx +
∫Ω
cT n+1qdx −∫
Ωτ(k grad T n+1, grad q)dx
=
∫Ω
cT nqdx +
∫Ωα div unqdx .
for all q ∈ H(Ω), v ∈ H(div,Ω).
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Implementation
The process of numerical solution of the problem:
Building geometry and mesh generation;Numerical implementation of the solver using Fenics library;Visualization of the results.
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Geometry and grids
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Numerical result. Temperature
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Numerical result. Displacement
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Numerical comparision of schemes with weight
Discrete-time form
µ∇2un+1 + (λ+ µ) grad div un+1 − α grad T n+1 = 0,
α div un+1 + cT n+1 − wτ div(k grad T n+1
)=
cT n + α div un + (1− w)τ div(k grad T n
).
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Split schemes
Solution strategies:Coupled schemesThe coupled governing equations oftemperature and mechanics are solvedsimultaneously at every time step.Split schemesEither the temperature, or mechanical,problem is solved first, and then theother problem is solved using theintermediate solution information.
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Split schemes
L-split (Drained split):freezing the temperature during solutionof the mechanic problem;uses updated mechanics when solvingtemperature problem.
Scheme
µ∇2un+1 + (λ+ µ) grad div un+1 − α grad T n = 0,
α div un+1 + cT n+1 − τ div(k grad T n+1
)= cT n + α div un.
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Split schemes
U-split (Fixed-strain scheme):freezing the displacement duringsolution of the temperature problem;uses updated temperature when solvingmechanic problem.
Scheme
µ∇2un+1 + (λ+ µ) grad div un+1 − α grad T n+1 = 0,
α div un + cT n+1 − τ div(k grad T n+1
)= cT n + α div un−1.
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Numerical comparision of split schemes
Schemes
Grids Time step
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Numerical comparision of schemes with weight
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Future works
Poroelastic problems
µ∇2u + (λ+ µ) grad div u − α grad p = 0,
α∂ div u∂t
+ S∂p∂t− div
(kµ
grad p)
= 0.
Elasto-plastic problems.
Thank you for your attention!