Updated lagrangian mixed finite element formulation for quasi and fully incompressible fluids

Abstract

We present a mixed velocity–pressure finite element formulation for solving the updated Lagrangian equations for quasi and fully incompressible fluids. Details of the governing equations for the conservation of momentum and mass are given in both differential and variational form. The finite element interpolation uses an equal order approximation for the velocity and pressure unknowns. The procedure for obtaining stabilized FEM solutions is outlined. The solution in time of the discretized governing conservation equations using an incremental iterative segregated scheme is described. The linearization of these equations and the derivation of the corresponding tangent stiffness matrices is detailed. Other iterative schemes for the direct computation of the nodal velocities and pressures at the updated configuration are presented. The advantages and disadvantages of choosing the current or the updated configuration as the reference configuration in the Lagrangian formulation are discussed.

Appendix: Linearization of the momentum equations with respect to the nodal velocities

Using the expression of ${}^{n+1}\bar{{\mathbf {r}}}_m$ of Eq. (62) and neglecting the changes of the external vector ${}^{n+1}\bar{{\mathbf {f}}}_m$ with the velocity (accounting for these changes is possible and will lead to additional terms in the tangent matrix), we can write

$$\begin{aligned}&\delta _{\bar{v}} {}^{n+1} \bar{{\mathbf {r}}}_m = \frac{1}{\Delta t}{{\mathbf {M}}}_v d \bar{{\mathbf {v}}}\nonumber \\&\quad + \int _{{}^{n}V} \Big \{ {{\mathbf {B}}}^T \Big [\delta _{\bar{v}}\left\{ \mathbf S ^{\prime } \right\} \left( \delta _{\bar{v}} J\left\{ {{\mathbf {C}}}^{-1}\right\} + {J} \delta _{\bar{v}} \left\{ {{\mathbf {C}}}^{-1}\right\} \right) p \nonumber \\&\quad + J \left\{ {{\mathbf {C}}}^{-1}\right\} \delta _{\bar{v}}p\Big ]+ \delta _{\bar{v}} {{\mathbf {B}}}^T \left\{ \mathbf S ^{\prime } \right\} \Big \}d{~}^{n} V \end{aligned}$$

(102)

Introducing Eqs. (55) and (59) into (102) gives after some algebra [4, 5, 36]

$$\begin{aligned}&\delta _{\bar{v}} \left\{ \mathbf S ' \right\} = [{\varvec{\mathcal {C}}}]\delta _{\bar{v}} \left\{ \dot{{\mathbf {E}}}\right\} = [{\varvec{\mathcal {C}}}] {{\mathbf {B}}} d \bar{{\mathbf {v}}}\end{aligned}$$

(103)

$$\begin{aligned}&\delta _{\bar{v}} J \left\{ {{\mathbf {C}}}^{-1}\right\} = J {{\mathbf {C}}}^{-1} \otimes {{\mathbf {C}}}^{-1} \delta _{\bar{v}} \left\{ {{\mathbf {E}}}\right\} = J \Delta t{{\mathbf {C}}}^{-1} \otimes {{\mathbf {C}}}^{-1} {{\mathbf {B}}} d \bar{{\mathbf {v}}}\end{aligned}$$

(104)

$$\begin{aligned}&J \delta _{\bar{v}} \left\{ {{\mathbf {C}}}^{-1}\right\} = - 2 J {\varvec{\mathcal {I}}} \delta _{\bar{v}} \left\{ {{\mathbf {E}}}\right\} = -2 J \Delta t {\varvec{\mathcal {I}}} {{\mathbf {B}}} d \bar{{\mathbf {v}}} \end{aligned}$$

(105)

where

$$\begin{aligned} {\varvec{\mathcal {I}}}_{ijkl}= \frac{1}{2} \left[ ({{\mathbf {C}}}^{-1})_{ik} ({{\mathbf {C}}}^{-1})_{jl}- ({{\mathbf {C}}}^{-1})_{il}({{\mathbf {C}}}^{-1})_{jk}\right] \end{aligned}$$

(106)

It can be shown that tensor ${\varvec{\mathcal {I}}}$ is symmetric [4, 5].

On the other hand,

$$\begin{aligned} \delta _{\bar{v}} {{\mathbf {B}}}^T \left\{ {{\mathbf {S}}}' \right\} = \Delta t {{\mathbf {G}}}^T \hat{{\mathbf {S}}}'{{\mathbf {G}}} d \bar{{\mathbf {v}}} \end{aligned}$$

(107a)

with

$$\begin{aligned} \begin{array}{lll} {{\mathbf {G}}}=\left[ \begin{array}{lll}\bar{{\mathbf {G}}} &{} \quad \bar{{\mathbf {0}}} &{} \quad \bar{{\mathbf {0}}}\\ \bar{{\mathbf {0}}} &{}\quad \bar{{\mathbf {G}}}&{}\quad \bar{{\mathbf {0}}}\\ \bar{{\mathbf {0}}} &{} \quad \bar{{\mathbf {0}}}&{} \quad \bar{{\mathbf {G}}} \end{array}\right] \, ,\, \\ \bar{{\mathbf {G}}} = \left[ \begin{array}{llllllll} {}_nN^v_{1,1} &{}\quad 0&{}\quad 0&{} \quad {}_nN^v_{2,1}&{}\quad 0&{}\quad 0&{} \quad \cdots &{} \quad {}_nN^v_{n,1}\\ {}_nN^v_{1,2} &{}\quad 0&{}\quad 0&{}\quad {}_nN^v_{2,2}&{}\quad 0&{}\quad 0&{} \quad \cdots &{} \quad {}_nN^v_{n,2}\\ {}_nN^v_{1,3} &{}\quad 0&{}\quad 0&{} \quad {}_nN^v_{2,3}&{}\quad 0&{}\quad 0&{} \quad \cdots &{} \quad {}_nN^v_{n,3} \end{array}\right] \\ \hat{{\mathbf {S}}}^{\prime } = \left[ \begin{array}{lll} {{\mathbf {S}}}^{\prime } &{} \quad {{\mathbf {0}}} &{} \quad {{\mathbf {0}}}\\ {{\mathbf {0}}} &{} \quad {{\mathbf {S}}}^{\prime }&{} \quad {{\mathbf {0}}}\\ {{\mathbf {0}}} &{} \quad {{\mathbf {0}}}&{} \quad {{\mathbf {S}}}^{\prime } \end{array}\right] , \, {{\mathbf {0}}} = \left[ \begin{array}{lll} 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0 \end{array}\right] ,\, \bar{{\mathbf {0}}} = \left\{ \begin{array}{l} 0\\ 0\\ 0 \end{array}\right\} \end{array} \nonumber \\ \end{aligned}$$

(107b)

with ${}_nN^v_{i,j}$ defined in Eq. (58b).

In the derivation of Eqs. (104), (105) and (107a) we have assumed that $d(\Delta u_{i})= du_{i} =\Delta t ~d ({}^{n+\alpha }v_{i}) = \Delta t ~dv_{i}$. This relationship follows from Eq. (60a).

Substituting Eqs. (103)–(105) and (107a) and the interpolation for the pressure (Eq. (53)) into (102) yields the linearized form of the residual vector of the discretized momentum equations as

$$\begin{aligned} \delta _v {}^{n+1}\bar{{\mathbf {r}}}_m&= \frac{1}{\Delta t}{{\mathbf {M}}}_v d \bar{{\mathbf {v}}} \nonumber \\&\quad + \left\{ \int _{{}^{n}V}\left[ {{\mathbf {B}}}^T [{\varvec{\mathcal {C}}}_T] {{\mathbf {B}}}+ {{\mathbf {G}}}^T \hat{{\mathbf {S}}}{{\mathbf {G}}}\right] d {~}^{n}V \right\} d \bar{{\mathbf {v}}} \nonumber \\&\quad +\left\{ \int _{{}^{n}V} {{\mathbf {B}}}^T \left\{ {{\mathbf {C}}}^{-1}\right\} {{\mathbf {N}}}_p J d {~}^{n}V \right\} {\delta }_{\bar{v}}^{n+1}\bar{{\mathbf {p}}}\nonumber \\&= \left( \frac{1}{\Delta t}{{\mathbf {M}}}_v + {{\mathbf {K}}}_c + {{\mathbf {K}}}_\sigma \right) d \bar{{\mathbf {v}}}+ {{\mathbf {Q}}} {\delta }_{\bar{v}}^{n+1}\bar{{\mathbf {p}}} \end{aligned}$$

(108)

where matrices ${{\mathbf {M}}}_v$ and ${{\mathbf {Q}}}$ were given in Eq. (67) and ${{\mathbf {K}}}_c$ and ${{\mathbf {K}}}_\sigma$ are the constitutive and initial stress matrices, respectively. The expression of these matrices is

$$\begin{aligned} {{\mathbf {K}}}_c = \!\!\int _{{}^{n}V}{{\mathbf {B}}}^T [{\varvec{\mathcal {C}}}_T] {{\mathbf {B}}} d {~}^{n}V ,\, {{\mathbf {K}}}_\sigma =\!\!\int _{{}^{n}V}{{\mathbf {G}}}^T \hat{{\mathbf {S}}}' {{\mathbf {G}}}d {~}^{n}V \end{aligned}$$

(109)

The tangent constitutive tensor ${\varvec{\mathcal {C}}}_T$ is deduced from Eqs. (102), (103)–(105) as

$$\begin{aligned} {\varvec{\mathcal {C}}}_T = {\varvec{\mathcal {C}}} + J \Delta t p ({{\mathbf {C}}}^{-1}\otimes {{\mathbf {C}}}^{-1} - 2 {\varvec{\mathcal {I}}}) \end{aligned}$$

(110)

It is straightforward to show that tensor ${\varvec{\mathcal {C}}}_T$ is symmetric.