Reduction of Interface Conditions

Author: T. Cluzeau(note: ), V. Dolean(note: ), F. Nataf(note: ), A. Quadrat(note: )


Table of Contents


1. Elasticity 2D
2. Elasticity 3D
3. Stokes 2D
4. Stokes 3D
5. Oseen 2D
     5.1. Case 1
          5.1.1. Correction step
          5.1.2. Update step
     5.2. Case 2
          5.2.1. Correction step
          5.2.2. Update step
     5.3. Case 3
          5.3.1. Correction step
          5.3.2. Update step
     5.4. Case 4
          5.4.1. Correction step
          5.4.2. Update step
6. Oseen 3D
     6.1. Case 1
          6.1.1. Correction step
          6.1.2. Update step
7. Links

1. Elasticity 2D

restart:with(linalg):with(OreModules):

We consider the system of the linear elasticity equations in two dimensions. This system is defined by Ry=0 with RA 2×2 a square matrix of size 2 with entries in the commutative Ore algebra A=Q(λ,μ)[dx,dy] of differential operators in dx,dy with coefficients in Q(λ,μ), where λ and μ are real parameters, dx=d/dx and dy=d/dy are the derivations with respect to x respectively y. We define the Ore algebra A and the system matrix R.

A:=DefineOreAlgebra(diff=[dx,x],diff=[dy,y],polynom=[x,y],comm=[lambda,mu]):

R :=evalm([[(2*mu+lambda)*dx^2+mu*dy^2,(lambda+mu)*dx*dy],[(lambda+mu)*dx*dy,(2*mu+lambda)*dy^2+mu*dx^2]]);

R :=2μ+λdx 2 +μdy 2 λ+μdxdyλ+μdxdy2μ+λdy 2 +μdx 2

The equations can be written:

G := convert(evalm(R&*[u,v]),set);

G:=2μ+λdx 2 +μdy 2 u+λ+μdxdyv,λ+μdxdyu+2μ+λdy 2 +μdx 2 v

We now define a new Ore algebra distinct from A in which dy is viewed as a parameter so that the only derivation involved is dx=d/dx. We also add u and v as new commuting variables.

B:=DefineOreAlgebra(diff=[dx,x],polynom=[u,v,x],comm=[dy,u,v,lambda,mu]):

We then define an appropriate term order to short the indeterminates of the Ore algebra B and compute a Gröbner basis of the set of equations G with respect to this term order.

mTord:=`OreModules/term_order`(B[1],tdeg(dx,u,v,x),[u,v]);

GB:=`OreModules/gb`(G,mTord);

GB:=[dxdyuλ+dxdyuμ+2vμdy 2 +vdy 2 λ+vμdx 2 ,2uμdx 2 +udx 2 λ+uμdy 2 +dxdyvλ+dxdyvμ]

We are now going to reduce the interface conditions with respect to the Groebner basis GB of the equations of the system. We first need to define the interface conditions and to achieve this, we compute the Smith normal form of R with respect to the variable dx.

S:=map(factor,smith(R,dx,´U´,´V´));

S:=100dy 2 +dx 2 2

The matrices U and V are unimodular matrices over the ring Q(λ,μ,dy)[dx] such that we have URV=S. Equivalently defining E=U 1 and F=V 1 , we have ESF=R. In our case, the matrices E and F are the following ones:

E:=inverse(U);F:=inverse(V);

E:=λ+μdxdyμ dy 2 2μdy 2 +dy 2 λ+μdx 2 μ 2 dx dy 3 λ+μ
F:=dy 2 λ+μdx 2 dx dy 3 λ+μ110

Given two operators Op 1 and Op 2, for example, Op 1=1 and Op 2=dx 2 +dy 2 , the interface conditions are computed from the matrix F (here only from the second row of F) as follows:

Op1:=1:IC_1:=Mult(Op1,linalg[submatrix](F,2..2,1..2),A);

IC 1 :=10

Op2:=dx^2+dy^2:IC_2:=Mult(Op2,linalg[submatrix](F,2..2,1..2),A);

IC 2 :=dy 2 +dx 2 0

We then reduce the interface conditions with respect to the Gröbner basis of the equations of the system. We take the first interface condition.

IC1:=evalm(IC_1&*evalm([[u],[v]]));

IC1:=u

Then we compute its normal form with respect to GB.

NIC1:=`OreModules/normal_form`(IC1[1,1],GB,mTord);

NIC1:=u

We then do the same with the second interface condition:

IC2:=evalm(IC_2&*evalm([[u],[v]]));

IC2:=dy 2 +dx 2 u

NIC2:=`OreModules/normal_form`(IC2[1,1],GB,mTord);

NIC2:=λ+μdxdyv 2μ+λ+dy 2 uλ+μ 2μ+λ

Finally, we perform linear algebra simplifications to the system formed by the two normal forms NIC1 and NIC2. To avoid any multiplication by dx in the computation, we first replace dxu and dxv by new jet variables u x and v x in NIC1 and NIC2 and subtract them by right hand sides f 1 and f 2 .

M1:=subs(u=u[x],v=v[x],coeff(NIC1,dx,1))+coeff(NIC1,dx,0)-f[1];

M1:=uf 1

M2:=subs(u=u[x],v=v[x],coeff(numer(NIC2),dx,1))+coeff(numer(NIC2),dx,0)-f[2];

M2:=λ+μdyv x +dy 2 uλ+μf 2

We finally solve {M1=0,M2=0} in the unknowns u x ,v x ,u,v and put f 1 =f 2 =0 in the result to obtain the reduced interface conditions.

Sols:=subs(f1=0,f2=0,solve(M1,M2,u[x],v[x],u,v));

Sols:=u=0,u x =u x ,v=v,v x =0

The reduced interface conditions found are thus

u(x,y)=0,v(x,y) x=0.

The result can be directly obtained using the procedure ReducedInterfaceConditions. The procedure takes as inputs the matrix R of the system, a unimodular matrix F corresponding to the Smith normal form of R (see explanations below), the Ore algebra A of the coefficients of R, the operators Op 1=1 and Op 2=dx 2 +dy 2 defining the interface conditions and the names of the variables u and v. The ouput contains the reduced interface conditions.

ReducedInterfaceConditions(R,F,A,[1,dx^2+dy^2],[u,v]);

[u=0,v x =0]

Given the matrix R, the unimodular matrices E and F such that ESF=R, where S is the Smith normal form of R, are not unique. Moreover, as we have seen before, the interface conditions considered depend on the choice of F so that we can obtained distinct reduced interface conditions for distinct choices of E and F. For example, here, another choice for F is

F1:=1dx3μdy 2 +2dy 2 λ+2μdx 2 +dx 2 λ dy 3 λ+μ01

from which we obtain the following reduced interface conditions:

ReducedInterfaceConditions(R,F1,A,[1,dx^2+dy^2],[u,v]);

[v=0,u x =0]

2. Elasticity 3D

restart:with(linalg):with(OreModules):

We consider the elastostatic equations (i.e., the Navier-Cauchy equations) in three dimensions. This system is defined by Ry=0 with RA 3×3 a square matrix of size 3 with entries in the commutative Ore algebra A=Q(λ,μ)[dx,dy,dz] of differential operators in dx,dy,dz with coefficients in Q(λ,μ) where λ and μ are real parameters, dx=d/dx, dy=d/dy and dz=d/dz are the derivations with respect to x respectively y and z. We define the Ore algebra A and the system matrix R.

A:=DefineOreAlgebra(diff=[dx,x],diff=[dy,y],diff=[dz,z],polynom=[x,y,z],comm=[lambda,mu]):

R := matrix(3, 3,[-2*dx^2*mu-dx^2*lambda-dy^2*mu-dz^2*mu,-dx*dy*(lambda+mu),-dx*dz*(lambda+mu),-dx*dy*(lambda+mu),-dx^2*mu-2*dy^2*mu-dy^2*lambda-dz^2*mu,-dy*dz*(lambda+mu),-dx*dz*(lambda+mu),-dy*dz*(lambda+mu),-dx^2*mu-dy^2*mu-2*dz^2*mu-dz^2*lambda]);

R:=2dx 2 μdx 2 λdy 2 μdz 2 μdxdyλ+μdxdzλ+μdxdyλ+μdx 2 μ2dy 2 μdy 2 λdz 2 μdydzλ+μdxdzλ+μdydzλ+μdx 2 μdy 2 μ2dz 2 μdz 2 λ

The Smith normal form of R with respect to the variable dx is given by:

S:=smith(R,dx,´U´,´V´);

S:=1000dx 2 +dy 2 +dz 2 000dy 4 +2dx 2 dy 2 +2dy 2 dz 2 +2dz 2 dx 2 +dx 4 +dz 4

The (reduced) interface conditions depend on the unimodular matrices E and F in Q(dy,dz,λ,μ) such that R=ESF. Let us first consider the F returned by the smith function of Maple.

F:=inverse(V);

F:=dx2dx 2 μ 2 +μdx 2 λ+dz 2 μ 2 dy 2 λ 2 2μdy 2 λ dyλ+μ2dy 2 μ+dy 2 λ+dz 2 μ1dzdx 2 μdy 2 λdy 2 μ dy2dy 2 μ+dy 2 λ+dz 2 μdxdz dy 2 +dz 2 01dy 2 dz 2 +dxdz dy 2 +dz 2 01

Choosing this particular F, the interface conditions are the following:

Delta:=dx^2+dy^2+dz^2:IC1:=Mult(dx,linalg[submatrix](F,2..2,1..3),A);IC2:=Mult(dx,linalg[submatrix](F,3..3,1..3),A);IC3:=Mult(dx*Delta,linalg[submatrix](F,3..3,1..3),A);

IC1:=dzdx 2 dy 2 +dz 2 0dx
IC2:=dxdy 2 dz 2 +dxdz dy 2 +dz 2 0dx
IC3:=dxdy 2 +dz 2 +dx 2 dy 2 dz 2 +dxdz dy 2 +dz 2 0dxdz 2 dxdy 2 dx 3

We reduce them using the ReducedInterfaceConditions procedure. This yields:

ReducedInterfaceConditions(R,F,A,[dx,dx,dx*Delta],[u,v,w]);

[v=dzw dy,u x =0,v x =w x dy 2 λ+dzudy 2 μ2w x dy 2 μ+udz 3 μw x dz 2 μ dydzλ+μ]

However, choosing others F provides distinct (simpler) reduced interface conditions. With

F1:=1dx3dy 2 μ+2dy 2 λ+2dx 2 μ+2dz 2 μ+dx 2 λ+dz 2 λ dydy 2 +dz 2 λ+μdxdz dy 2 +dz 2 0dzdy001

we obtain

ReducedInterfaceConditions(R,F1,A,[dx,dx,dx*Delta],[u,v,w]);

[u=0,v x =0,w x =0]

and with

F2:=1dx3dy 2 μ+2dy 2 λ+2dx 2 μ+2dz 2 μ+dx 2 λ+dz 2 λ dydy 2 +dz 2 λ+μdxdz dy 2 +dz 2 0dzdy100

we get

ReducedInterfaceConditions(R,F2,A,[dx,dx,dx*Delta],[u,v,w]);

[v=dzw dy,u x =0,v x =dyw x dz]

3. Stokes 2D

restart:with(linalg):with(OreModules):

We consider the Stokes equations in two dimensions. This system is defined by Ry=0 with RA 3×3 a square matrix of size 3 with entries in the commutative Ore algebra A=Q(ν,c)[dx,dy] of differential operators in dx,dy with coefficients in Q(ν,c), where ν and c are real parameters, dx=d/dx and dy=d/dy are the derivations with respect to x respectively y. We define the Ore algebra A and the system matrix R.

A:=DefineOreAlgebra(diff=[dx,x],diff=[dy,y],polynom=[x,y],comm=[nu,c]):

R:=evalm([[-nu*(dx^2+dy^2)+c,0,dx],[0,-nu*(dx^2+dy^2)+c,dy],[dx,dy,0]]);

R:=νdx 2 +dy 2 +c0dx0νdx 2 +dy 2 +cdydxdy0

The Smith normal form of R with respect to the variable dx is given by:

S:=smith(R,dx,´U´,´V´);

S:=100010002dy 2 νdx 2 dy 4 ν+dy 2 cνdx 4 +dx 2 c ν

The (reduced) interface conditions depend on the unimodular matrices E and F in Q(dy,dz,λ,μ) such that R=ESF. Let us consider for example the F returned by the smith function of Maple.

F:=inverse(V);

F:=0νdx 2 νdy 2 +c dy1dx dy10100

Choosing this particular F, the interface conditions are the following:

L:=-nu*(dx^2+dy^2)+c:IC1:=Mult(dx,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(dx*L,linalg[submatrix](F,3..3,1..3),A);

IC1:=dx00
IC2:=νdx 3 dxνdy 2 +dxc00

We reduce them using the ReducedInterfaceConditions procedure. This yields:

ReducedInterfaceConditions(R,F,A,[dx,dx*L],[u,v,p]);

[p=0,v=0]

4. Stokes 3D

restart:with(linalg):with(OreModules):

We consider the Stokes equations in three dimensions. This system is defined by Ry=0 with RA 4×4 a square matrix of size 4 with entries in the commutative Ore algebra A=Q(ν,c)[dx,dy,dz] of differential operators in dx,dy,dz with coefficients in Q(ν,c), where ν and c are real parameters, dx=d/dx, dy=d/dy and dz=d/dz are the derivations with respect to x respectively y and z. We define the Ore algebra A and the system matrix R.

A:=DefineOreAlgebra(diff=[dx,x],diff=[dy,y],diff=[dz,z],polynom=[x,y,z],comm=[nu,c]):

R:=evalm([[-nu*(dx^2+dy^2+dz^2)+c,0,0,dx],[0,-nu*(dx^2+dy^2+dz^2)+c,0,dy],[0,0,-nu*(dx^2+dy^2+dz^2)+c,dz],[dx,dy,dz,0]]);

R:=νdx 2 +dy 2 +dz 2 +c00dx0νdx 2 +dy 2 +dz 2 +c0dy00νdx 2 +dy 2 +dz 2 +cdzdxdydz0

The Smith normal form of R with respect to the variable dx is given by:

S:=smith(R,dx,´U´,´V´);

S:=1000010000dx 2 νdy 2 νdz 2 +c ν0000νdx 2 νdy 2 νdz 2 +cdx 2 +dy 2 +dz 2 ν

The (reduced) interface conditions depend on the unimodular matrices E and F in Q(dy,dz,λ,μ) such that R=ESF. Let us consider for example the F returned by the smith function of Maple.

F:=inverse(V);

F:=0νdx 2 νdy 2 νdz 2 +c dy01dx dy1dz dy0dzdx dy 2 +dz 2 010dy 2 dz 2 +dzdx dy 2 +dz 2 010

Choosing this particular F, the interface conditions are the following:

L:=-nu*(dx^2+dy^2+dz^2)+c:IC1:=Mult(dx,linalg[submatrix](F,3..3,1..4),A);IC2:=Mult(dx,linalg[submatrix](F,4..4,1..4),A);IC3:=Mult(dx*L,linalg[submatrix](F,4..4,1..4),A);

IC1:=dzdx 2 dy 2 +dz 2 0dx0
IC2:=dxdy 2 dz 2 +dxdz dy 2 +dz 2 0dx0
IC3:=dy 2 dz 2 +dxdzdxνdy 2 νdz 2 νdx 2 +c dy 2 +dz 2 0dxνdy 2 +dxνdz 2 +νdx 3 dxc0

We reduce them using the ReducedInterfaceConditions procedure. This yields:

ReducedInterfaceConditions(R,F,A,[dx,dx,L*dx],[u,v,w,p]);

[p=0,v=dzw dy,p x =dzuνdy 2 uνdz 3 +dzuc+w x νdy 2 +w x νdz 2 dz]

5. Oseen 2D

restart:with(linalg):with(OreModules):

We consider the Oseen equations in two dimensions. This system is defined by Ry=0 with RA 3×3 a square matrix of size 3 with entries in the commutative Ore algebra A=Q(ν,c,b1,b2)[dx,dy] of differential operators in dx,dy with coefficients in Q(ν,c,b1,b2), where ν,c,b1,b2 are real parameters, dx=d/dx and dy=d/dy are the derivations with respect to x respectively y. We define the Ore algebra A and the system matrix R.

A:=DefineOreAlgebra(diff=[dx,x],diff=[dy,y],polynom=[x,y],comm=[nu,c,b1,b2]):

R:=evalm([[-nu*(dx^2+dy^2)+c+b1*dx+b2*dy,0,dx],[0,-nu*(dx^2+dy^2)+c+b1*dx+b2*dy,dy],[dx,dy,0]]);

R:=νdx 2 +dy 2 +c+b1dx+b2dy0dx0νdx 2 +dy 2 +c+b1dx+b2dydydxdy0

The Smith normal form of R with respect to the variable dx is given by:

S:=map(factor,smith(R,dx,´U´,´V´));

S:=10001000dx 2 +dy 2 νdx 2 +b1dxνdy 2 +c+b2dy ν

The (reduced) interface conditions depend on the unimodular matrices E and F in Q(dy,dz,λ,μ) such that R=ESF. Let us consider for example the F returned by the smith function of Maple.

F:=inverse(V);

F:=0νdx 2 +b1dxνdy 2 +c+b2dy dy1dx dy10100

Choosing this particular F, we compute the reduced interface conditions using the ReducedInterfaceConditions procedure.

Four cases of interface conditions have to be distinguished and reduced. To write these four cases, we introduce the following differential operators:

Delta:=dx^2+dy^2:L2:=-nu*(dx^2+dy^2)+b1*dx+b2*dy+c:Robin:=nu*dx-b1/2:

5.1. Case 1

5.1.1. Correction step

The interface conditions are the following:

IC1:=Mult(Robin,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(L2*dx,linalg[submatrix](F,3..3,1..3),A);

IC1:=νdx1/2b100
IC2:=νdx 3 +b1dx 2 dxνdy 2 +dxc+dxb2dy00

We reduce them:

ReducedInterfaceConditions(R,F,A,[Robin,L2*dx],[u,v,p]);

[p=0,u=2νdyv b1]

5.1.2. Update step

The interface conditions are the following:

IC1:=Mult(L2,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(1,linalg[submatrix](F,3..3,1..3),A);

IC1:=νdx 2 +b1dxνdy 2 +c+b2dy00
IC2:=100

We reduce them:

ReducedInterfaceConditions(R,F,A,[L2,1],[u,v,p]);

[u=0,p x =0]

5.2. Case 2

5.2.1. Correction step

The interface conditions are the following:

IC1:=Mult(Robin*Delta,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(dx,linalg[submatrix](F,3..3,1..3),A);

IC1:=νdx 3 +dxνdy 2 1/2b1dx 2 1/2dy 2 b100
IC2:=dx00

We reduce them:

ReducedInterfaceConditions(R,F,A,[Robin*Delta,dx],[u,v,p]);

[v=0,p x =2νdy 2 p+b1ub2dy2νudy 2 b1+b1uc b1]

5.2.2. Update step

The interface conditions are the following:

IC1:=Mult(1,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(L2,linalg[submatrix](F,3..3,1..3),A);

IC1:=100
IC2:=νdx 2 +b1dxνdy 2 +c+b2dy00

We reduce them:

ReducedInterfaceConditions(R,F,A,[1,L2],[u,v,p]);

[u=0,p x =0]

5.3. Case 3

5.3.1. Correction step

The interface conditions are the following:

IC1:=Mult(dx*L2,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(1,linalg[submatrix](F,3..3,1..3),A);

IC1:=νdx 3 +b1dx 2 dxνdy 2 +dxc+dxb2dy00
IC2:=100

We reduce them:

ReducedInterfaceConditions(R,F,A,[dx*L2,1],[u,v,p]);

[p=0,u=0]

5.3.2. Update step

The interface conditions are the following:

IC1:=Mult(L2,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(Robin,linalg[submatrix](F,3..3,1..3),A);

IC1:=νdx 2 +b1dxνdy 2 +c+b2dy00
IC2:=νdx1/2b100

We reduce them:

ReducedInterfaceConditions(R,F,A,[L2,Robin],[u,v,p]);

[u=2νdyv b1,p x =0]

5.4. Case 4

5.4.1. Correction step

The interface conditions are the following:

IC1:=Mult(Delta*Robin,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(1,linalg[submatrix](F,3..3,1..3),A);

IC1:=νdx 3 +dxνdy 2 1/2b1dx 2 1/2dy 2 b100
IC2:=100

We reduce them:

ReducedInterfaceConditions(R,F,A,[Delta*Robin,1],[u,v,p]);

[u=0,p x =dy2νdyp+2νvc+2νdyvb2+b1 2 v b1]

5.4.2. Update step

The interface conditions are the following:

IC1:=Mult(Delta,linalg[submatrix](F,3..3,1..3),A);IC2:=Mult(dx,linalg[submatrix](F,3..3,1..3),A);

IC1:=dx 2 +dy 2 00
IC2:=dx00

We reduce them:

ReducedInterfaceConditions(R,F,A,[Delta,dx],[u,v,p]);

[u=p x c+b2dy,v=0]

6. Oseen 3D

restart:with(linalg):with(OreModules):

We consider the Oseen equations in three dimensions. This system is defined by Ry=0 with RA 4×4 a square matrix of size 4 with entries in the commutative Ore algebra A=Q(ν,c,b1,b2,b3)[dx,dy,dz] of differential operators in dx,dy,dz with coefficients in Q(ν,c,b1,b2), where ν,c,b1,b2,b3 are real parameters, dx=d/dx,dy=d/dy and dz=d/dz are the derivations with respect to x respectively y and z. We define the Ore algebra A and the system matrix R.

A:=DefineOreAlgebra(diff=[dx,x],diff=[dy,y],diff=[dz,z],polynom=[x,y,z],comm=[nu,c,b1,b2,b3]):

L:=-nu*dx^2-nu*dy^2-nu*dz^2+b1*dx+b2*dy+b3*dz+c;R:=evalm([[L,0,0,dx],[0,L,0,dy],[0,0,L,dz],[dx,dy,dz,0]]);

L:=νdx 2 +b1dxνdy 2 νdz 2 +c+b2dy+b3dz
R:=L00dx0L0dy00Ldzdxdydz0

The Smith normal form of R with respect to the variable dx is given by:

S:=map(factor,smith(R,dx,´U´,´V´));

S:=1000010000L ν0000Ldx 2 +dy 2 +dz 2 ν

The (reduced) interface conditions depend on the unimodular matrices E and F in Q(dy,dz,λ,μ) such that R=ESF. Let us consider for example the F returned by the smith function of Maple.

F:=inverse(V);

F:=0νdx 2 +b1dxνdy 2 νdz 2 +c+b2dy+b3dz dy01dx dy1dz dy0dzdx dy 2 +dz 2 010dy 2 dz 2 +dzdx dy 2 +dz 2 010

Choosing this particular F, we compute the reduced interface conditions using the ReducedInterfaceConditions procedure.

Four cases of interface conditions have to be distinguished and reduced. Here, we will only give the computed reduced interface conditions for the first case: the other ones can be obtained in the same way but we then get huge expressions that are not really readable. To write these four cases, we define the following differential operators:

Delta:=dx^2+dy^2+dz^2:Robin:=nu*dx-b1/2:

6.1. Case 1

6.1.1. Correction step

The interface conditions are the following:

IC1:=Mult(Robin,linalg[submatrix](F,3..3,1..4),A);IC2:=Mult(Robin,linalg[submatrix](F,4..4,1..4),A);IC3:=Mult(dx*L,linalg[submatrix](F,4..4,1..4),A);

IC1:=1/22νdx+b1dzdx dy 2 +dz 2 0νdx1/2b10
IC2:=1/22νdx+b1dy 2 dz 2 +dxdz dy 2 +dz 2 0νdx+1/2b10
IC3:=dy 2 dz 2 +dxdzdxνdx 2 νdy 2 νdz 2 +b1dx+b2dy+b3dz+c dy 2 +dz 2 0dxL0

We reduce them:

ReducedInterfaceConditions(R,F,A,[Robin,Robin,dx*L],[u,v,w,p]);

[p=0,u=2νdyv b12νdzw b1,p x =2ν 2 dy 3 v b12ν 2 wdzdy 2 b1+2νdy 2 vb2 b1+1/2b1 2 w2νw x b1dy 2 b1dz2ν 2 dyvdz 2 b1+1/24b2νw+4νb3vdzdy b1+1/2vb1 2 +4νcvdy b12ν 2 wdz 3 b1+2b3νwdz 2 b1+1/22b1 2 w2νw x b1+4cνwdz b1]

6.1.2. Update step

The interface conditions are the following:

IC1:=Mult(1,linalg[submatrix](F,3..3,1..4),A);IC2:=Mult(L,linalg[submatrix](F,4..4,1..4),A);IC3:=Mult(1,linalg[submatrix](F,4..4,1..4),A);

IC1:=dxdz dy 2 +dz 2 010
IC2:=νdx 2 νdy 2 νdz 2 +b1dx+b2dy+b3dz+cdy 2 dz 2 +dxdz dy 2 +dz 2 0L0
IC3:=dy 2 dz 2 +dxdz dy 2 +dz 2 010

We reduce them:

ReducedInterfaceConditions(R,F,A,[1,L,1],[u,v,w,p]);

[u=0,w=dzv dy,p x =0]

7. Links

The computations have been done using the Maple package OreModules(note: ).

This web page has been produced by the LaTeX to xml converter Tralics(note: ).

Notes


Note 1. University of Limoges ; CNRS ; XLIM UMR 6172 ; DMI 123 avenue Albert Thomas, 87060 Limoges cedex, France, cluzeau@ensil.unilim.fr


Note 2. University of Nice Sophia-Antipolis, Laboratoire J.A.Dieudonné, UMR CNRS 6621, Parc Valrose 06108 NICE Cedex 2, France, victorita.dolean@unice.fr


Note 3. , Université Pierre et Marie Curie, Laboratoire J.L. Lions, 175 rue du Chevaleret 75013 Paris, France, nataf@ann.jussieu.fr


Note 4. INRIA Sophia Antipolis, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France, Alban.Quadrat@sophia.inria.fr


Note 5. http://wwwb.math.rwth-aachen.de/OreModules/


Note 6. http://www-sop.inria.fr/miaou/tralics/