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Q-Tensor Gradient Flow with Quasi-Entropy and Discretizations Preserving Physical Constraints

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Abstract

We propose and analyze numerical schemes for the gradient flow of Q-tensor with the quasi-entropy. The quasi-entropy is a strictly convex, rotationally invariant elementary function, giving a singular potential constraining the eigenvalues of Q within the physical range \((-1/3,2/3)\). Compared with the potential derived from the Bingham distribution, the quasi-entropy has the same asymptotic behavior and underlying physics. Meanwhile, it is very easy to evaluate because of its simple expression. For the elastic energy, we include all the rotationally invariant terms. The numerical schemes for the gradient flow are built on the nice properties of the quasi-entropy. The first-order time discretization is uniquely solvable, keeping the physical constraints and energy dissipation, which are all independent of the time step. The second-order time discretization keeps the first two properties unconditionally, and the third with an O(1) restriction on the time step. These results also hold when we further incorporate a second-order discretization in space. Error estimates are also established for time discretization and full discretization. Numerical examples about defect patterns are presented to validate the theoretical results.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

Yanli Wang is partially supported by NSFC Nos. 12171026, U1930402 and 12031013. Jie Xu is partially supported by NSFC Nos. 12288201 and 12001524.

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JX contributed to the conceptualization, methodology, analysis and first draft. YW contributed to the coding, data processing and visualization. Both authors revised and approved the final manuscript.

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Correspondence to Jie Xu.

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Appendices

A Bingham distribution and its properties

We briefly introduce the Bingham term mentioned in the main text. Detailed discussions could be found in the literature (see, for example, [4]).

For \(Q\in {\mathcal {Q}}_{\textrm{phys}}\), consider the minimization problem

$$\begin{aligned}&\min \int _{S^2}\rho (\varvec{m})\ln \rho (\varvec{m})\,\textrm{d}\varvec{m},\\&\mathrm{s.t. }\quad \int _{S^2}\rho (\varvec{m})\,\textrm{d}\varvec{m}=1,\quad \int _{S^2}\left( \varvec{m}\otimes \varvec{m}-\frac{1}{3}I\right) \rho (\varvec{m})\,\textrm{d}\varvec{m}=Q. \end{aligned}$$

This problem has a unique solution of the form

$$\begin{aligned} \rho (\varvec{m})=\frac{1}{Z}\exp \Big (B(Q)\cdot (\varvec{m}\otimes \varvec{m}-\frac{1}{3}I)\Big ),\qquad Z=\int _{S^2}\exp \Big (B(Q)\cdot (\varvec{m}\otimes \varvec{m}-\frac{1}{3}I)\Big )\,\textrm{d}\varvec{m}, \end{aligned}$$
(A.1)

where B(Q) is a symmetric traceless \(3\times 3\) matrix uniquely determined by Q. Such a density function is the Bingham distribution. The Bingham term in the free energy is given by taking (A.1) into \(\int _{S^2}\rho \ln \rho \,\textrm{d}\varvec{m}\), giving

$$\begin{aligned} \psi (Q)=B(Q)\cdot Q-\ln Z. \end{aligned}$$
(A.2)

It is not difficult to show that \(\psi (Q)\) is rotationally invariant and strictly convex. As a result, we could focus on the case where Q is diagonal, which implies that B(Q) is also diagonal. Let us denote \(Q=\textrm{diag}(\lambda _1,\lambda _2,\lambda _3)\) and \(B(Q)=\textrm{diag}(\mu _1,\mu _2,\mu _3)\). If we assume \(\lambda _1\geqslant \lambda _2\geqslant \lambda _3\), then it holds \(\mu _1\geqslant \mu _2\geqslant \mu _3\). The asymptotic behavior of \(\min \lambda (Q)\rightarrow (-1/3)^+\) is exactly that of \(\mu _1-\mu _3\rightarrow +\infty \). It is shown in [4] that when \(\mu _1-\mu _3\rightarrow +\infty \),

$$\begin{aligned}&\lambda _3+\frac{1}{3}=\frac{C_1}{\mu _1-\mu _3}+o\left( \frac{1}{\mu _1-\mu _3}\right) ,\\&\psi (Q)=C_2\ln (\mu _1-\mu _3)+o(\mu _1-\mu _3). \end{aligned}$$

Therefore, \(\psi (Q)\sim -C_3\ln (\lambda _3+1/3)\).

It is noticed that to compute (A.2), it is necessary to solve B(Q) first. However, the existing numerical approaches [17, 22,23,24, 32] are still not able to provide a fast and accurate evaluation of B(Q).

B Summation by parts

We derive the first equality in (3.31b). Substituting (3.28) into (3.31), we deduce that that

$$\begin{aligned} \sum _{P} (u D_{12}v)_{P} = -\frac{1}{4h^2} \sum _{l,m=1}^{N-1} u_{l,m} (-v_{l+1, m+1} + v_{l+1, m-1} + v_{l-1, m+1} - v_{l-1, m-1}).\qquad \quad \end{aligned}$$
(B.1)

Substituting (3.27) into (B.1), we derive that

$$\begin{aligned}&\sum _{\Gamma }(D_1 u)_{\Gamma }(D_{2}v)_{\Gamma }\nonumber \\ =&\frac{1}{4h^2} \sum _{l=0}^{N-1}\sum _{m=0}^{N-1}(-u_{l,m} -u_{l,m+1} + u_{l+1, m} + u_{l+1, m+1}) (-v_{l,m}+v_{l,m+1}-v_{l+1, m}+v_{l+1, m+1}) \nonumber \\ =&\frac{1}{4h^2} \sum _{l=0}^{N-1}\sum _{m=0}^{N-1}u_{l,m} (v_{l,m}-v_{l,m+1}+v_{l+1,m}-v_{l+1, m+1}) \nonumber \\&+\frac{1}{4h^2} \sum _{l=0}^{N-1}\sum _{m=1}^{N}u_{l,m} (v_{l,m-1}-v_{l,m}+v_{l+1, m-1}-v_{l+1, m}) \nonumber \\&+ \frac{1}{4h^2} \sum _{l=1}^{N}\sum _{m=0}^{N-1}u_{l,m} (-v_{l-1,m}+v_{l-1,m+1}-v_{l, m}+v_{l, m+1}) \nonumber \\&+ \frac{1}{4h^2} \sum _{l=1}^{N}\sum _{m=1}^{N}u_{l,m} (-v_{l-1,m-1}+v_{l-1,m}-v_{l, m-1}+v_{l, m}) \nonumber \\ =&\frac{1}{4h^2} \sum _{l=1}^{N-1}\sum _{m=1}^{N-1}u_{l,m} (-v_{l+1, m+1} + v_{l+1, m-1} + v_{l-1, m+1} - v_{l-1, m-1})\nonumber \\&+ \frac{1}{4h^2}\left( F_1(u,v)_{0,0} + \sum _{m=1}^{N-1}F_1(u,v)_{0,m} + \sum _{l=1}^{N-1}F_1(u,v)_{l,0}\right) \nonumber \\&+ \frac{1}{4h^2}\left( F_2(u,v)_{0,N} + \sum _{m=1}^{N-1}F_2(u,v)_{0,m} + \sum _{l=1}^{N-1}F_2(u,v)_{l,N}\right) \nonumber \\&+ \frac{1}{4h^2}\left( F_3(u,v)_{N,0} + \sum _{m=1}^{N-1}F_3(u,v)_{N,m} + \sum _{l=1}^{N-1}F_3(u,v)_{l,0}\right) \nonumber \\&+ \frac{1}{4h^2}\left( F_4(u,v)_{N,N} + \sum _{m=1}^{N-1}F_4(u,v)_{N,m} + \sum _{l=1}^{N-1}F_4(u,v)_{l,N}\right) , \end{aligned}$$
(B.2)

with

$$\begin{aligned} \begin{aligned} F_1(u, v)_{l,m}&= u_{l,m}(v_{l,m}-v_{l,m+1}+v_{l+1, m}-v_{l+1, m+1}),\\ F_2(u, v)_{l,m}&= u_{l,m}(v_{l,m-1}-v_{l,m}+v_{l+1, m-1}-v_{l+1, m}), \\ F_3(u, v)_{l,m}&= u_{l,m}(-v_{l-1,m}+v_{l-1,m+1}-v_{l, m}+v_{l, m+1}), \\ F_4(u, v)_{l, m}&= u_{l,m}(-v_{l-1,m-1}+v_{l-1,m}-v_{l, m-1}+v_{l, m}). \end{aligned} \end{aligned}$$
(B.3)

In each \(F_i(u,v)\) in (B.2), the indices are located on the boundary. Since u is zero on boundary nodes, the first equality in (3.31b) has already been established.

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Wang, Y., Xu, J. Q-Tensor Gradient Flow with Quasi-Entropy and Discretizations Preserving Physical Constraints. J Sci Comput 94, 9 (2023). https://doi.org/10.1007/s10915-022-02060-x

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