R3iCe

R3iCe formulation

Install

Tutorial : run simulation

Simulation output

R3iCe formulation

R3iCe formulation

Numerical implementation

Description of the numerical algorithm

System of equation

P1 i.

\[\tilde{S}=\tilde{\eta}^\star\left(2\alpha_1 \tilde{D} + 2\alpha_2 M^D Tr(M\tilde{D}) + \alpha_3 (M\tilde{D}+\tilde{D}M)^D\right)\]

where

\[\tilde{\eta}^\star=2\left(\alpha_1 tr(\tilde{D}^2) + \alpha_2 tr(M\tilde{D})^2+\alpha_3 tr(M\tilde{D}^2)\right)^{\frac{1-n}{2n}}\]

P1 ii.

\[-\tilde{\nabla}\tilde{S}+\tilde{\nabla}\tilde{p}=0\]

P2.

\[\dot{\tilde{c}}=\mathbf{\tilde{W}}(\tilde{u}).c-\lambda \left[\mathbf{\tilde{D}}(\tilde{u}).c-(c^T.\mathbf{\tilde{D}}(\tilde{u}).c).c \right] + \mathcal{Mo} (c_0-c)\]

Time discretization

The time discretization is using a BDF2 differentiation formula where \(\Delta \tilde{t}\) is the discrete time step.

P1 i.

\[\tilde{S}^{t+1}={\tilde{\eta}^{t+1}}^\star\left(2\alpha_1 \tilde{D}(u^{t+1}) + 2\alpha_2 {M^{t+1}}^D Tr(M^{t+1}\tilde{D}(u^{t+1})) + \alpha_3 (M^{t+1}\tilde{D}(u^{t+1})+\tilde{D}(u^{t+1})M^{t+1})^D\right)\]

where

\[{\tilde{\eta}^{t+1}}^\star = 2\left(\alpha_1 tr({\tilde{D}(u^{t+1})}^2) + \alpha_2 tr(M^{t+1}\tilde{D}(u^{t+1}))^2+\alpha_3 tr(M^{t+1}{\tilde{D}(u^{t+1})}^2)\right)^{\frac{1-n}{2n}}\]

P1 ii.

\[-\tilde{\nabla}\tilde{S}^{t+1}+\tilde{\nabla}\tilde{p}^{t+1}=0\]

P2.

\[\frac{1}{\Delta \tilde{t}}\left(c^{t+1}-\frac{4}{3}c^t+\frac{1}{3}c^{t-1}\right)=\frac{2}{3}\left(\mathbf{\tilde{W}}(\tilde{u}^{t+1}).c^{t+1}-\lambda \left[\mathbf{\tilde{D}}(\tilde{u}^{t+1}).c^{t+1}-({c^{t+1}}^T.\mathbf{\tilde{D}}(\tilde{u^{t+1}}).c^{t+1}).c^{t+1} \right] + \mathcal{Mo} (c_0^{t+1}-c^{t+1})\right)\]

Numerical algorithm

The numerical algorithm is based on using two nested fixed loops. The first loop solves the non-linear formulation of the CTI law, while the second loop is used for the time step. To ensure the algorithm converges quickly, it is recommended to limit the maximum number of iterations for the non-linear fixed loop to one. The fixed point is included in the code to enable R3iCe simulation with non-linear CTI without time evolution.

main program

initialization

• generate \(c^0\)

• \(c^{-1}=c^{-2}=c^{-0}\)

• \(u^0=0\)

• \(p^0=0\)

time loop : find \(c^{t+1},~u^{t+1},~p^{t+1}\) knowing \(c^{t},~c^ {t-1}\)

fix point to solve P2 \(k_c\)

initialization

• \(c^{t+1,k_c=0}=c^{t}\)

• \(u^{t+1,k_c=0}=c^{t}\)

Fix point to converge to \(c^{t+1}\) and \(u^{t+1}\)

fix point to solve non linear P1 \(k_{nl}\)

initialization

• \(u^{t+1,k_c,k_{nl}=0}=u^{t+1,k_c}\)

Fix point to converge to \(u^{t+1,k_c}\) with \(c^{t+1,k_c}\) fix

rheolef magic solve linearized P1

Find \(u^{t+1,k_c,k_{nl}+1}\) solving P1 with

• \(c^{t+1,k_c}\)

• \({\tilde{\eta}^\star}^{t+1,k_c,k_{nl}}(c^{t+1,k_c},u^{t+1,k_c,k_{nl}})\)

Update :

• \({\tilde{\eta}^\star}^{t+1,k_c,k_{nl}+1}(c^{t+1,k_c},u^{t+1,k_c,k_{nl}+1})\)

Compute residual :

\[res_{nl} = | S(c^{t+1,k_c},u^{t+1,k_c,k_{nl}+1}) - S(c^{t+1, k_c},u^{t+1,k_c,k_{nl}}) |\]

if \(res_{nl}<tol_{nl}\) or \(k_{nl}> it\_max\_nl\) : STOP

\(u^{t+1,k_c+1}=u^{t+1,k_c,k_{nl}}\)

Find \(c_0(u^{t+1,k_c+1})\) and compute :

\(c^{t+1,k_c+1}(u^{t+1,k_c},c^{t},c^{t-1},c_0)\) using P2

Compute residual :

\[res_{c} = | S(c^{t+1,k_c+1},u^{t+1,k_c+1}) - S(c^{t+1,k_c},u^{t+1,k_c+1}) |\]

if \(res_{c}<tol_{c}\) or \(k_{c} > it\_max\_c\) : STOP

\(u^{t+1,k_c+1}=u^{t+1,k_c,k_{nl}}\)

convergence successful

\(c^{t+1}\) and \(u^{t+1}\) found

do \(nb\_t\) time loop