On the derivation of a component-free scheme for Lagrangian fluid–structure interaction problems

Abstract

The main goal of this work is the resolution of fluid structure interaction problems described with the Lagrangian formalism by means of a consistently derived monolithic approach. The use of a component-free derivation leads to a straightforward implementation of the formulation where only vectors and second-order tensors in ${\mathbb {R}}^3$ are required. Therefore, no basis or components have to be imposed ab initio for the discrete variational formulation as occurs when Voigt notation is employed. The computational framework adopted is the local maximum-entropy material point method (LME-MPM), a mesh-free technique that combines the material point sampling of the MPM and the LME, a spatial approximation technique with basis functions of class $C^{\infty }$. This framework sidesteps the use of expensive mesh refinement techniques, which are typically required when Lagrangian finite element method is employed. Finally, the effectiveness of this approach is illustrated against challenging fluid dynamic problems.

Appendices

Appendix A Auxiliary derivations

Lemma 1

Let $\delta {\varphi }$ be the variation of the differentiable mapping ${\varphi }$; then, the linearization of the deformation gradient ${{\textbf{F}}}$ is given by:

$$\begin{aligned} \delta {{\textbf{F}}}\ =\ \nabla _{0}{D{\varphi }}. \end{aligned}$$

(A1)

Proof of Lemma 1

The linearization of the deformation gradient has the following structure:

$$\begin{aligned} \frac{d}{d\epsilon } \Bigg |_{\epsilon = 0}\ {{\textbf{F}}}({\varphi }_{n+1} + \epsilon D{\varphi })\ =\ \frac{\partial ({\varphi }_{n+1} + \epsilon D{\varphi })}{\partial {\textbf{X}}}\ =\ \nabla _{0}{D{\varphi }}. \end{aligned}$$

(A2)

$\square$

Lemma 2

The linearization of ${{\textbf{F}}}^{-1}$ reads as:

$$\begin{aligned} \delta {{\textbf{F}}}^{-1}\ =\ - {{\textbf{F}}}^{-1}\ \nabla _{0}{D{\varphi }}\ {{\textbf{F}}}^{-1}. \end{aligned}$$

(A3)

Remark

An immediate result of Lemma 2 is:

$$\begin{aligned} \delta {{\textbf{F}}}^{-T}\ =\ - {{\textbf{F}}}^{-T}\ \nabla _{0}{D{\varphi }}^{T}\ {{\textbf{F}}}^{-T}. \end{aligned}$$

(A4)

Proof of Lemma 2

Consider the relation ${\textbf{A}}^{-1}{\textbf{A}} = {{\textbf{I}}}$ as starting point:

$$\begin{aligned} \begin{aligned}&\delta {\textbf{A}}^{-1}\ {\textbf{A}}\ +\ {\textbf{A}}^{-1}\ \delta {\textbf{A}}\ =\ \delta {{\textbf{I}}}\ =\ 0\\&\delta {\textbf{A}}^{-1}\ {\textbf{A}}\ =\ - {\textbf{A}}^{-1}\ \delta {\textbf{A}}\\&\delta {\textbf{A}}^{-1}\ =\ - {\textbf{A}}^{-1}\ \delta {\textbf{A}}\ {\textbf{A}}^{-1} \end{aligned} \end{aligned}$$

(A5)

Therefore, the variation of the inverse for any non-singular tensor ${\textbf{A}}$ can be expressed as Eq. (A5). $\square$

Lemma 3

Let J be the determinant, or Jacobian, of the deformation gradient ${{\textbf{F}}}$ and ${{\textbf{F}}^*}$ the cofactor of ${{\textbf{F}}}$. Then, the linearization of J is:

$$\begin{aligned} \delta {J} = {{\textbf{F}}^*}: \nabla _{0}{D{\varphi }}. \end{aligned}$$

(A6)

Proof of Lemma 3

For a ${\mathbb {R}}^3 \times {\mathbb {R}}^3$ non-singular matrix ${\textbf{A}}$, i.e., one with nonzero determinant, the Cayley–Hamilton theorem states the following characteristic equation:

$$\begin{aligned} p({\textbf{A}})\ =\ {\textbf{A}}^3\ -\ tr({\textbf{A}}){\textbf{A}}^2\ +\ I_{2}{\textbf{A}}\ -\ det({\textbf{A}}){{\textbf{I}}}\ = {\textbf{0}}. \end{aligned}$$

(A7)

$I_{2}$ is the second invariant of ${\textbf{A}}$. For the particular case of ${{\textbf{F}}}$ and rearranging the expression of the characteristic equation:

$$\begin{aligned} {J}\ {{\textbf{F}}}^{-1} = {{\textbf{F}}}^2 - tr({{\textbf{F}}}){{\textbf{F}}} + I_{2}{{\textbf{I}}}. \end{aligned}$$

Considering the result obtained in Lemma 1:

$$\begin{aligned} \begin{aligned} \frac{d}{d\epsilon } \Bigg |_{\epsilon = 0}\ {J}({\varphi }_{n+1} + \epsilon D{\varphi })&= \frac{\partial {J}}{\partial {{\textbf{F}}}}: \frac{d}{d\epsilon } \Bigg |_{\epsilon = 0}\ {{\textbf{F}}}({\varphi }_{n+1} + \epsilon D{\varphi })\\&= ({{\textbf{F}}}^2 - tr({{\textbf{F}}}){{\textbf{F}}} + I_2{{\textbf{I}}})^T: \nabla _{0}{D{\varphi }}\\&= {J} {{\textbf{F}}}^{-T}: \nabla _{0}{D{\varphi }}\\&= {{\textbf{F}}^*}: \nabla _{0}{D{\varphi }}. \end{aligned} \end{aligned}$$

$\square$

Lemma 4

The linearization of the cofactor of the deformation gradient ${{\textbf{F}}}$ is:

$$\begin{aligned} \delta {{\textbf{F}}^*}\ =\ \left[ ({{\textbf{F}}}^{-T}: \nabla _{0}{D{\varphi }}){{\textbf{I}}}\ -\ {{\textbf{F}}}^{-T}\nabla _{0}{D{\varphi }}^T \right] {{\textbf{F}}^*}. \end{aligned}$$

(A8)

Proof of Lemma 4

Following the definition of ${{\textbf{F}}^*}$, applying the chain rule over it and considering the intermediate results in Lemmas 1 and 3:

$$\begin{aligned} \begin{aligned} \delta {{\textbf{F}}^*}&= \delta {J}{{\textbf{F}}}^{-T}\ +\ {J}\delta {{\textbf{F}}}^{-T} \\&= {J} \left( {{\textbf{F}}}^{-T}: \nabla _{0}{D{\varphi }} \right) {{\textbf{F}}}^{-T}\ -\ {J}{{\textbf{F}}}^{-T}\ \nabla _{0}{D{\varphi }}^{T}\ {{\textbf{F}}}^{-T}\\&= \left[ ({{\textbf{F}}}^{-T}: \nabla _{0}{D{\varphi }}){{\textbf{I}}}\ -\ {{\textbf{F}}}^{-T}\nabla _{0}{D{\varphi }}^T \right] {{\textbf{F}}^*} \end{aligned} \end{aligned}$$

$\square$

Lemma 5

The linearization of the rate of the deformation gradient ${\dot{{\textbf{F}}}}$ is:

$$\begin{aligned} \delta {\dot{{\textbf{F}}}}\ = \ \alpha _4\nabla _{0}{D{\varphi }}. \end{aligned}$$

(A9)

Proof of Lemma 5

Taking as starting point the linearization of the deformation gradient obtained in Lemma 1 and having in mind the time discretization presented in Sect. 3.2:

$$\begin{aligned} \begin{aligned} \frac{d}{d\epsilon } \Bigg |_{\epsilon = 0}\ {\dot{{\textbf{F}}}}({\varphi }_{n+1} + \epsilon D{\varphi })&= \frac{d}{d\epsilon } \Bigg |_{\epsilon = 0}\ \alpha _4 \left( {\varphi }_{n + 1} + \epsilon D{\varphi } - {\varphi }_{n} \right) + \alpha _5 {{\textbf {v}}}_{n} + \alpha _6 {{\textbf {a}}}_{n}\ \\&= \alpha _4\nabla _{0}{D{\varphi }}. \end{aligned} \end{aligned}$$

$\square$

Lemma 6

The linearization of the material velocity gradient $\nabla {{{\textbf {v}}}}$ is:

$$\begin{aligned} \delta \nabla {{{\textbf {v}}}}\ =\ \left( \alpha _4{{\textbf{I}}}\ -\ \nabla {{{\textbf {v}}}} \right) \nabla _{0}{D{\varphi }}{{\textbf{F}}}^{-1}. \end{aligned}$$

(A10)

Remark

By the application of Lemma 6, the linearization of ${{\textbf{d}}}$ is:

$$\begin{aligned} \delta {{\textbf{d}}}\ =\ \frac{1}{2}\ \left( \left( \alpha _4{{\textbf{I}}}\ -\ \nabla {{{\textbf {v}}}} \right) \nabla _{0}{D{\varphi }}{{\textbf{F}}}^{-1}\ +\ {{\textbf{F}}}^{-T}\nabla _{0}{D{\varphi }}^T\left( \alpha _4{{\textbf{I}}}\ -\ \nabla {{{\textbf {v}}}^T} \right) \right) . \end{aligned}$$

(A11)

Proof of Lemma 6

Considering the definition of $\nabla {{{\textbf {v}}}}$ as in Eq. (5) and applying the intermediate results obtained in Lemma 2 and Lemma 5:

$$\begin{aligned} \begin{aligned} \delta \nabla {{{\textbf {v}}}}&= \delta {\dot{{\textbf{F}}}}{{\textbf{F}}}^{-1}\ +\ {\dot{{\textbf{F}}}}\delta {{\textbf{F}}}^{-1}\\&= \alpha _4\nabla _{0}{D{\varphi }}{{\textbf{F}}}^{-1}\ -\ {\dot{{\textbf{F}}}}{{\textbf{F}}}^{-1}\nabla _{0}{D{\varphi }}{{\textbf{F}}}^{-1}\\&=\ \left( \alpha _4{{\textbf{I}}}\ -\ \nabla {{{\textbf {v}}}} \right) \nabla _{0}{D{\varphi }}{{\textbf{F}}}^{-1}. \end{aligned} \end{aligned}$$

$\square$

Lemma 7

For any vectors ${{\textbf {v}}},{{\textbf {b}}},\delta {{\textbf {a}}} \in {\mathbb {R}}^{dim}$ and any fourth-order tensor ${\mathbb {T}}: \square$, the following relation is satisfied:

$$\begin{aligned} {\mathbb {T}} \{{{\textbf {v}}},{{\textbf {b}}} \}\ \delta {{\textbf {a}}}\ = \left[ {\mathbb {T}}: (\delta {{\textbf {a}}} \otimes {{\textbf {b}}}) \right] {{\textbf {v}}}. \end{aligned}$$

(A12)

This leads to the definition of the operator:

$$\begin{aligned} {\mathbb {T}} \{{\mathbb {R}}^{dim},{\mathbb {R}}^{dim}\}\ \rightarrow \ {\mathbb {R}}^{dim} \times {\mathbb {R}}^{dim}. \end{aligned}$$

(A13)

From this definition, it is a simple exercise to find the bilinear tensor form ${\mathbb {T}} \{\square ,\square \}$ for the various simple, but frequent, tensor-valued linear operations collected in Table 7.

Table 7 Expression of the tensor-valued bilinear form ${\mathbb {T}} \{{{\textbf {v}}},{{\textbf {b}}} \}$ for simple linear tensor transformations. Naming $\delta {{\textbf {a}}} \otimes {{\textbf {b}}}$ as the displacement-gradient tensor $\delta {{\textbf {H}}}$

Remark

Consider as example of application ${{\textbf {v}}} = \nabla _{0}{N({{\textbf {x}}})}^{\beta }$, ${{\textbf {b}}} = \nabla _{0}{N({{\textbf {x}}})}^{\alpha }$, $\delta {{\textbf {a}}} = D{\varphi }_{n+1}^{\beta }$, and ${\mathbb {T}}: \square = {\mathbb {A}}: \square$. We pretend to factorize out the variation of the displacement, $D{\varphi }^{\beta }_{n+1}$, in the foregoing expression:

$$\begin{aligned} \begin{aligned}&\Big [{\psi }^{\alpha } \otimes \nabla _{0}{N({\textbf{x}})^{\alpha }}\Big ]\,\ \Big [{\mathbb {A}}: \left[ D{\varphi }_{n+1}^{\beta } \otimes \nabla _{0}{N({\textbf{x}})^{\beta }} \right] \Big ]\\&\quad = {\psi }^{\alpha }\ \cdot \Big [{\mathbb {A}}: \left[ D{\varphi }_{n+1}^{\beta } \otimes \nabla _{0}{N({\textbf{x}})^{\beta }} \right] \Big ]\ \cdot \ \nabla _{0}{N({\textbf{x}})^{\alpha }}. \end{aligned} \end{aligned}$$

(A14)

By the application of Lemma 7:

$$\begin{aligned} {\psi }^{\alpha }\ \cdot \ {\mathbb {A}} \big \{ \nabla _{0}{N({\textbf{x}})^{\alpha }},\ \nabla _{0}{N({\textbf{x}})^{\beta }} \big \}\ \cdot \ D{\varphi }^{I}_{n+1}. \end{aligned}$$

(A15)

Proof of Lemma 7

See Planas et al. [63]. $\square$

Appendix B Linearization of the variational equation for the solid phase

The component-free expression of the tangent density ${\mathbb {A}} \big \{\square ,\square \big \}$ for a general isotropic hyperelastic material whose strain energy density function W is function of the principal invariants of ${{\textbf{C}}}$ is given by Stickle et al. [73]:

$$\begin{aligned} \begin{aligned}&{\mathbb {A}} \big \{\nabla _{0}{N({\textbf{x}})^{\alpha }},\ \nabla _{0}{N({\textbf{x}})^{\beta }} \big \}\ \\&\quad = {{\textbf{S}}}: (\nabla _{0}{N({\textbf{x}})^{\alpha }} \otimes \nabla _{0}{N({\textbf{x}})^{\beta }}){{\textbf{I}}}\ +\ {{\textbf{F}}} {\mathbb {C}} \big \{ \nabla _{0}{N({\textbf{x}})^{\alpha }},\ \nabla _{0}{N({\textbf{x}})^{\beta }} \big \} {{\textbf{F}}}^T, \end{aligned} \end{aligned}$$

(B16)

where

$$\begin{aligned} {\mathbb {C}} \big \{{{\textbf {v}}},{{\textbf {b}}}\big \}= & {} \Gamma _1 ({{\textbf{C}}}^{-1}{{\textbf {v}}}\ \otimes \ {{\textbf{C}}}^{-1}{{\textbf {b}}})\ +\ \Gamma _2 ({{\textbf {v}}}\ \otimes \ {{\textbf{C}}}^{-1}{{\textbf {b}}})\nonumber \\{} & {} + \Gamma _2 ({{\textbf{C}}}^{-1}{{\textbf {v}}}\ \otimes \ {{\textbf {b}}})\ +\ \Gamma _3 ({{\textbf{C}}}{{\textbf {v}}}\ \otimes \ {{\textbf{C}}}^{-1}{{\textbf {b}}})\nonumber \\{} & {} + \Gamma _3 ({{\textbf{C}}}^{-1}{{\textbf {v}}}\ \otimes \ {{\textbf{C}}}{{\textbf {b}}})\ +\ \Gamma _4 ({{\textbf {v}}}\ \otimes \ {{\textbf {b}}})\nonumber \\{} & {} + \Gamma _5 ({{\textbf {v}}}\ \otimes \ {{\textbf{C}}}{{\textbf {b}}})\ +\ \Gamma _5 ({{\textbf{C}}}{{\textbf {v}}}\ \otimes \ {{\textbf {b}}})\nonumber \\{} & {} + \frac{\Gamma _7}{2}( ({{\textbf {v}}} \cdot {{\textbf {b}}}){{\textbf{I}}}\ +\ ({{\textbf {v}}}\ \otimes \ {{\textbf {b}}}))\nonumber \\{} & {} + \frac{\Gamma _8}{2}(({{\textbf{C}}}^{-1}{{\textbf {b}}} \cdot {{\textbf {v}}}){{\textbf{C}}}^{-1}\ +\ ({{\textbf{C}}}^{-1}{{\textbf {b}}}\ \otimes \ {{\textbf{C}}}^{-1}{{\textbf {v}}})). \end{aligned}$$

(B17)

The coefficients $\Gamma _i$ are scalar functions of the principal invariants of ${{\textbf{C}}}$, see Doghri [21]. The full development of Eq. (B17) has been omitted for the sake of clarity. Interested readers are recommended to refer to Stickle et al. [73] for the detailed derivation. For the particular case of a Neo-Hookean material, these coefficients $\Gamma _i$ are:

$$\begin{aligned} \begin{aligned} \Gamma _1&= \Lambda {J}^2, \quad \Gamma _8\ =\ -\Lambda ({J}^2 - 1)\ +\ 2G\\ \Gamma _i&= 0 \quad \text {for} \quad i\ =\ 2,\ldots ,7. \end{aligned} \end{aligned}$$

(B18)