ElasticNitscheMatrix

Interface1

This routine computes the following

$$\begin{aligned}\int_{\Gamma}\left(\delta{\bf u}\cdot{\bf e}\right)\left(\sigma:{\bf e}\otimes{\bf n}\right)dS & =\int_{\Gamma}\left(\delta{\bf u}\cdot{\bf e}\right)\left(\sigma:{\bf e}\otimes{\bf n}\right)dS\\ & =\int_{\Gamma}\left(\delta{\bf u}\cdot{\bf e}\right)\left\{ \lambda\left(\nabla\cdot{\bf u}\right)\left({\bf n}\cdot{\bf e}\right)+\mu\left({\bf u}\otimes\nabla\right):\left({\bf e}\otimes{\bf n}+{\bf n}\otimes{\bf e}\right)\right\} \\ & =\delta u_{iI}\left[{\bf M}\right]_{IJ}^{ij}u_{jJ} \end{aligned}$$

where, we have used

$$\sigma_{ij}n_{j}=\lambda\left(\frac{\partial u_{k}}{\partial x_{k}}\right)n_{i}+\mu\frac{\partial u_{i}}{\partial x_{k}}n_{k}+\mu\frac{\partial u_{k}}{\partial x_{i}}n_{k}$$

In the following routine lambda, mu, and evec are the FEVariable_.

INTERFACE
  MODULE PURE FUNCTION ElasticNitscheMatrix(Test, Trial, Lambda, Mu, Evec) &
    & RESULT(Ans)
    CLASS(ElemshapeData_), INTENT(IN) :: Test, Trial
    CLASS(FEVariable_), INTENT(IN) :: Lambda, Mu, Evec
    REAL(DFP), ALLOCATABLE :: Ans(:, :)
  END FUNCTION ElasticNitscheMatrix
END INTERFACE

In the following routine lambda, mu are constant and real values, and Evec is FEVariable_.

INTERFACE
  MODULE PURE FUNCTION ElasticNitscheMatrix(Test, Trial, Lambda, Mu, Evec) &
    & RESULT(Ans)
    CLASS(ElemshapeData_), INTENT(IN) :: Test, Trial
    CLASS(FEVariable_), INTENT(IN) :: Evec
    REAL(DFP), INTENT(IN) :: Lambda, Mu
    REAL(DFP), ALLOCATABLE :: Ans(:, :)
  END FUNCTION ElasticNitscheMatrix
END INTERFACE

Interface2

$$\begin{aligned}-\int_{\Gamma_{f}}\delta u_{i}\sigma_{ij}n_{j}dS & =-\delta u_{iI}\int_{\Gamma_{f}}N^{I}\sigma_{ij}n_{j}dS\\ & =-\delta u_{iI}\int_{\Gamma_{f}}N^{I}\lambda\frac{\partial u_{j}}{\partial x_{j}}n_{i}dS\\ & -\delta u_{iI}\int_{\Gamma_{f}}N^{I}\mu\frac{\partial u_{i}}{\partial x_{k}}n_{k}dS\\ & -\delta u_{iI}\int_{\Gamma_{f}}N^{I}\mu\frac{\partial u_{j}}{\partial x_{i}}n_{j}dS \end{aligned}$$ $$\begin{aligned}-\int_{\Gamma_{f}}\delta u_{i}\sigma_{ij}n_{j}dS & =-\delta u_{iI}\left[\int_{\Gamma_{f}}N^{I}\lambda\frac{\partial N^{J}}{\partial x_{j}}n_{i}dS\right]u_{jJ}\\ & -\delta u_{iI}\left[\int_{\Gamma_{f}}N^{I}\mu\frac{\partial N^{I}}{\partial x_{k}}n_{k}dS\right]u_{iI}\\ & -\delta u_{iI}\left[\int_{\Gamma_{f}}N^{I}\mu\frac{\partial N^{J}}{\partial x_{i}}n_{j}dS\right]u_{jJ} \end{aligned}$$ $$-\int_{\Gamma_{f}}\delta u_{i}\sigma_{ij}n_{j}dS=-\delta u_{iI}\left[{\bf N}_{1}\right]_{IJ}^{ij}u_{jJ}$$ $$\left[{\bf N}_{1}\right]_{IJ}^{ij}=\int_{\Gamma_{f}}N^{I}\lambda\frac{\partial N^{J}}{\partial x_{j}}n_{i}dS+\delta_{ij}\int_{\Gamma_{f}}N^{I}\mu\frac{\partial N^{I}}{\partial x_{k}}n_{k}dS+\int_{\Gamma_{f}}N^{I}\mu\frac{\partial N^{J}}{\partial x_{i}}n_{j}dS$$

In the following routine lambda, mu are instances of FEVariable_.

INTERFACE
  MODULE PURE FUNCTION ElasticNitscheMatrix(Test, Trial, Lambda, Mu, isNoSlip)&
    & RESULT(Ans)
    CLASS(ElemshapeData_), INTENT(IN) :: Test, Trial
    CLASS(FEVariable_), INTENT(IN) :: Lambda, Mu
    LOGICAL(LGT), INTENT(IN) :: isNoSlip
    REAL(DFP), ALLOCATABLE :: Ans(:, :)
  END FUNCTION ElasticNitscheMatrix
END INTERFACE

In the following routine lambda and mu are constant and real values.

INTERFACE
  MODULE PURE FUNCTION ElasticNitscheMatrix(Test, Trial, Lambda, Mu, isNoSlip)&
    & RESULT(Ans)
    CLASS(ElemshapeData_), INTENT(IN) :: Test, Trial
    REAL(DFP), INTENT(IN) :: Lambda, Mu
    LOGICAL(LGT), INTENT(IN) :: isNoSlip
    REAL(DFP), ALLOCATABLE :: Ans(:, :)
  END FUNCTION ElasticNitscheMatrix
END INTERFACE

Interface3

$$\left[{\bf N}_{n3}\right]_{IJ}^{ij}=\int_{\Gamma_{f}}\alpha N^{I}e_{i}N^{J}e_{j}$$

INTERFACE
  MODULE PURE FUNCTION ElasticNitscheMatrix(Test, Trial, Alpha, Evec) &
    & RESULT(Ans)
    CLASS(ElemshapeData_), INTENT(IN) :: Test, Trial
    CLASS(FEVariable_), INTENT(IN) :: Alpha, Evec
    REAL(DFP), ALLOCATABLE :: Ans(:, :)
  END FUNCTION ElasticNitscheMatrix
END INTERFACE

INTERFACE
  MODULE PURE FUNCTION ElasticNitscheMatrix(Test, Trial, Alpha, Evec) &
    & RESULT(Ans)
    CLASS(ElemshapeData_), INTENT(IN) :: Test, Trial
    CLASS(FEVariable_), INTENT(IN) :: Evec
    REAL(DFP), INTENT(IN) :: Alpha
    REAL(DFP), ALLOCATABLE :: Ans(:, :)
  END FUNCTION ElasticNitscheMatrix
END INTERFACE