Computational algorithms for thermoelastic problem

Petr N. Vabishchevich, Maria V. Vasilyeva

SCTEMMJuly 8-10, 2013, Yakutsk

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 .

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 .

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,

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.

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).

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.

Geometry and grids

Numerical result. Temperature

Numerical result. Displacement

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

).

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.

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.

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.

Numerical comparision of split schemes

Schemes

Grids Time step

Numerical comparision of schemes with weight

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!