Finite Elasticity model

[stoykov]

Dear All,

I am looking in the development of the Finite Elasticity model and the applied linearization (for the Newton's method), and I wish to ask if someone can confirm or correct my understanding of the code.
In the definition of the LocalMatrix at file ElasticSolve.src, the force vestor is defined in the following way:

ForceVector(dim*(p-1)+i) = ForceVector(dim*(p-1)+i) &
+(Basis(p)*Force(i)*DetDefG &
+Basis(p)*InertialForce(i)*Density &
-DOT_PRODUCT(dBasisdx(p,:),Stress1(i,:)) &
+DOT_PRODUCT(dBasisdx(p,:),dStress1U(i,:)))*s

May I assume that the term DOT_PRODUCT(dBasisdx(p,:),Stress1(i,:)) presents the internal force vector? And the term DOT_PRODUCT(dBasisdx(p,:),dStress1U(i,:)) presents the vector which is result of multiplication of the jacobian and the dispalcement vector?
The stiffness matrix is defined in this way:

StiffMatrix(dim*(p-1)+i,dim*(q-1)+j) &
= StiffMatrix(dim*(p-1)+i,dim*(q-1)+j) &
+ DOT_PRODUCT(dBasisdx(q,:),dStress1(j,:))*s

Does it mean that DOT_PRODUCT(dBasisdx(q,:),dStress1(j,:)) is the jacobian of the internal force vector?

Regards,
Stan

[mika]

Hi,

Concerning the RHS

- the part DOT_PRODUCT(dBasisdx(p,:),Stress1(i,:)) corresponds to the integration of the inner product <-S(F_k),grad v)> where S(F_k) gives the first Piola-Kirchhoff stress in the current state with the deformation gradient F_k and v is the test function
- the part DOT_PRODUCT(dBasisdx(p,:),Stress1(i,:)) corresponds to the integration of the inner product <DS(F_k)[grad u_k],grad v)> where DS(F) is the derivative of the first Piola-Kirchhoff stress at F

Conserning the stiffness matrix

- the part DOT_PRODUCT(dBasisdx(q,:),dStress1(j,:)) corresponds to the inner product <DS(F_k)[grad u_{k+1}],grad v)> where DS(F) is again the derivative of the first Piola-Kirchhoff stress at F

For further explanation see also the material around the Eq. (6.7) in the Elmer Models Manual at

http://www.nic.funet.fi/pub/sci/physics ... Manual.pdf

Best regards,
Mika