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A biologically-inspired mesh optimizer based on pseudo-material remodeling


Moving boundaries and interfaces are commonly encountered in fluid flow simulations. For instance, fluid-structure interaction simulations require the formulation of the problem in moving domains, making the mesh distortion an issue of concern towards ensuring the accuracy of numerical model predictions. In this work, we propose a technique for the simultaneous mesh optimization and motion characterization. The mesh optimization/motion method introduced here is inspired by the mechanobiology of soft tissues, particularly those present in arterial walls, which feature an incredible capability to adapt to altered mechanical stimuli through adaptive mechanisms such as growth and remodeling. The proposed approach is in the framework of a low-distortion mesh moving method that is based on fiber-reinforced hyperelasticity and optimized zero-stress state. We adopt different reference configurations for the different constituents, namely ground substance and fibers. Hypothetical reference configurations are postulated for the different pieces of pseudo-material (the elements) as target shapes. Also, we modify the equilibrium equations using a volume-invariant strategy. Through the introduction of growth and remodeling adaptive processes we build an optimization algorithm which can attain an optimal configuration through a series of consecutive nonlinear optimizations steps. The remodeling mechanism allows to adapt the fiber deposition orientations, which become the driving force towards an homeostatic state, that is the optimal configuration. Also, a recruitment mechanism is introduced to selectively deal with initial highly distorted elements where high stresses develop due to the departure from the ideal configuration. We report 2D and 3D numerical experiments to show the application of this biologically-inspired mesh optimizer (BIMO) to simplicial finite element meshes. We also present additional numerical tests using BIMO as a mesh moving method. The results show that the proposed method performs satisfactorily, either as mesh optimizer and/or mesh motion strategy.

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This work was partially supported (N. Biocca, J.M. Gimenez and G.E. Carr) by CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas), Argentina. P.J. Blanco acknowledges the financial support of the Brazilian agencies CNPq (grant number 407751/2018-1) and FAPESP (grant number 2014/50889-7). The support of these agencies is gratefully acknowledged.

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A Basics relations

Consider a displacement field \({\mathbf {u}}({\mathbf {x}}_{t_n})\) which is perturbed producing \({\mathbf {u}}_{\epsilon } = {\mathbf {u}}+ \epsilon \delta {\mathbf {u}}\). With this perturbation we have the deformation gradient, originally given by \({\mathbf {F}}^{{\mathbf {u}}}= {\mathbf {I}} + \nabla {\mathbf {u}}\), results in \({\mathbf {F}}^{{\mathbf {u}}}_{\epsilon } = {\mathbf {I}} + \nabla {\mathbf {u}}_{\epsilon } = {\mathbf {I}} + \nabla ({\mathbf {u}}+ \epsilon \delta {\mathbf {u}})\). Let us compute the expressions of derivatives of several quantities involving \({\mathbf {F}}^{{\mathbf {u}}}_{\epsilon }\) with respect to \(\epsilon \). This will be employed in the linearization procedure whenever the configuration \(\varOmega _{t_{n}}\) is known. Then, the following expressions arise

$$\begin{aligned}&\frac{d}{d\epsilon } {\mathbf {F}}^{{\mathbf {u}}}_{\epsilon } \bigg |_{\epsilon =0} = \nabla \delta {\mathbf {u}}, \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } {\mathbf {F}}^{{\mathbf {u}}{\mathbf {v}}}_{\epsilon } \bigg |_{\epsilon =0} = \nabla \delta {\mathbf {u}}\,{\mathbf {F}}^{{\mathbf {v}}}, \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } ({\mathbf {F}}^{{\mathbf {u}}}_{\epsilon })^{-1} \bigg |_{\epsilon =0} = -{{\mathbf {F}}^{{\mathbf {u}}}}^{-1} \nabla \delta {\mathbf {u}}\,{{\mathbf {F}}^{{\mathbf {u}}}}^{-1}, \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } ({\mathbf {F}}^{{\mathbf {u}}{\mathbf {v}}}_{\epsilon })^{-1} \bigg |_{\epsilon =0} = -{{\mathbf {F}}^{{\mathbf {u}}{\mathbf {v}}}}^{-1} \nabla \delta {\mathbf {u}}\,{\mathbf {F}}^{{\mathbf {v}}}{{\mathbf {F}}^{{\mathbf {u}}{\mathbf {v}}}}^{-1}, \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } J^{\mathbf {u}}_{\epsilon } \bigg |_{\epsilon =0} = J^{\mathbf {u}}\left( {{\mathbf {F}}^{{\mathbf {u}}}}^{-T} \cdot \nabla \delta {\mathbf {u}}\right) , \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } J^{{\mathbf {u}}{\mathbf {v}}}_{\epsilon } \bigg |_{\epsilon =0} = J^{{\mathbf {u}}{\mathbf {v}}} \left( {{\mathbf {F}}^{{\mathbf {u}}}}^{-T} \cdot \nabla \delta {\mathbf {u}}\right) , \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } {\mathbf {E}}^{{\mathbf {u}}}_{\epsilon } \bigg |_{\epsilon =0} \nonumber \\&\quad = \left( {{\mathbf {F}}^{{\mathbf {u}}}}^T\nabla \delta {\mathbf {u}}\right) ^S, \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } {\mathbf {E}}^{{\mathbf {u}}{\mathbf {v}}}_{\epsilon } \bigg |_{\epsilon =0} \nonumber \\&\quad = \left( {{\mathbf {F}}^{{\mathbf {u}}{\mathbf {v}}}}^T \nabla \delta {\mathbf {u}}\,{\mathbf {F}}^{{\mathbf {v}}}\right) ^S, \end{aligned}$$
$$\begin{aligned}&\quad \frac{d}{d\epsilon } \left( \frac{J^R}{J^M_{\epsilon }}\right) ^{\chi }\bigg |_{\epsilon =0} \nonumber \\&\quad = -\chi \left( \frac{J^R}{J^M}\right) ^{\chi } \left( {{\mathbf {F}}^{{\mathbf {u}}}}^{-T} \cdot \nabla \delta {\mathbf {u}}\right) . \end{aligned}$$

Linearization procedure

The linearization of the variational expression given by Eq. (27) is presented. We recall that for this case \(\varOmega _{t_{n}}\) is fixed and for each Newton-Raphson iteration a new configuration \(\varOmega _{t_{n+1}}^{k}\) is obtained, with coordinates \({\mathbf {x}}_{t_{n+1}}^{k} = {\mathbf {x}}_{t_n} + {\mathbf {u}}_{n+1}^{k}\). Moreover, the virtual configuration \(\varOmega _{v_n}\) is fixed and the virtual displacement \({\mathbf {v}}_{n}\) as well. For the sake of clarity, we omit the index \(n+1\) on the displacement field \({\mathbf {u}}\) (and also on their related quantities, such as \({\mathbf {F}}^{{\mathbf {u}}}\)). Likewise we omit the index n on the virtual field \({\mathbf {v}}\) (and also on their related quantities, such as \({\mathbf {F}}^{{\mathbf {v}}}\)).

In abstract form the variational expression given by Eq. (27) reads: find \({\mathbf {u}}\in {\mathcal {U}}_{t_{n}}\) such

$$\begin{aligned} \langle {\mathcal {R}}_{t_n} ({\mathbf {u}}),{\hat{{\mathbf {u}}}}\rangle _{\varOmega _{t_{n}}} = 0 \quad \forall {\hat{{\mathbf {u}}}}\in {\mathcal {V}}_{t_{n}}. \end{aligned}$$

The Newton-Raphson linearization applied to the above expression at the point \({\mathbf {u}}^k \in {\mathcal {U}}_{t_n}\) and the increment \(\delta {\mathbf {u}}\in {\mathcal {V}}_{t_n}\) gives

$$\begin{aligned} \langle {\mathcal {R}}_{t_n} ({\mathbf {u}}^k),{\hat{{\mathbf {u}}}}\rangle _{\varOmega _{t_{n}}} + \frac{d}{d\epsilon } \langle {\mathcal {R}}_{t_n} ({\mathbf {u}}^k+\epsilon \delta {\mathbf {u}}),{\hat{{\mathbf {u}}}}\rangle _{\varOmega _{t_{n}}} \Big |_{\epsilon =0} = 0 \quad \forall {\hat{{\mathbf {u}}}}\in {\mathcal {V}}_{t_{n}}. \end{aligned}$$

As shown in Appendix A, to introduce the perturbation \(\epsilon \delta {\mathbf {u}}\) into quantities that depend on field \({\mathbf {u}}\), we introduce the additional index \(\epsilon \), for instance, \({\mathbf {F}}^{{\mathbf {u}}}_\epsilon = {\mathbf {I}} + \nabla ({\mathbf {u}}+ \epsilon \delta {\mathbf {u}}) = {\mathbf {F}}^{{\mathbf {u}}}+ \epsilon \nabla \delta {\mathbf {u}}\). As previously, we omit index k indicating quantities evaluated at the previous iteration. Thus, the perturbed residual (second term) from Eq. (43) takes the form

$$\begin{aligned}&\frac{d}{d\epsilon } \langle {\mathcal {R}}_{t_n} ({\mathbf {u}}^k+\epsilon \delta {\mathbf {u}}),{\hat{{\mathbf {u}}}}\rangle _{\varOmega _{t_{n}}} \Big |_{\epsilon =0} \nonumber \\&\quad =\int _{\varOmega _{t_{n}}} \dfrac{d}{d\epsilon } \left[ \left( {\mathbf {S}}_{g,\epsilon } + \frac{1}{J^{\mathbf {v}}} {\mathbf {F}}^{{\mathbf {v}}}{\mathbf {S}}_{f,\epsilon } {{\mathbf {F}}^{{\mathbf {v}}}}^T \right) \cdot {\dot{{\mathbf {E}}}}_{\epsilon }({\hat{{\mathbf {u}}}}) \left( \frac{J^{R}}{J^{M}_{\epsilon }} \right) ^\chi \right] \Bigg |_{\epsilon =0} d\varOmega _{t_{n}}, \end{aligned}$$

where \({\dot{{\mathbf {E}}}}_{\epsilon }({\hat{{\mathbf {u}}}}) = \frac{1}{2}\left( ({\mathbf {F}}^{{\mathbf {u}}}_{\epsilon })^T (\nabla {\hat{{\mathbf {u}}}}) + (\nabla {\hat{{\mathbf {u}}}})^T {\mathbf {F}}^{{\mathbf {u}}}_{\epsilon }\right) \), \({\mathbf {S}}_{g,\epsilon }={\mathbf {S}}_g({\mathbf {E}}^{{\mathbf {u}}}_{\epsilon })\) and \({\mathbf {S}}_{f,\epsilon } ={\mathbf {S}}_f({\mathbf {E}}^{{\mathbf {u}}{\mathbf {v}}}_{\epsilon })\) are the perturbed second Piola-Kirchhoff tensors (see Eqs. (15) and (16)), and \(J^M_{\epsilon } = J^M_n J^{\mathbf {u}}_{\epsilon }\) (with \(J^{M}_{n} = \text {det}(\partial {\mathbf {x}}_{t_{n}}/\partial \xi )\)) is the mesh Jacobian. Using expressions in Appendix A the derivation with respect to \(\epsilon \) (and posterior evaluation at \(\epsilon = 0\)) yields

$$\begin{aligned}&\frac{d}{d\epsilon } \langle {\mathcal {R}}_{t_n} ({\mathbf {u}}+\epsilon \delta {\mathbf {u}}),{\hat{{\mathbf {u}}}}\rangle _{\varOmega _{t_{n}}} \Big |_{\epsilon =0} \nonumber \\&\quad = \int _{\varOmega _{t_{n}}} \left( \frac{\partial {\mathbf {S}}_g}{\partial {\mathbf {E}}^{{\mathbf {u}}}} \left( {\nabla \delta {\mathbf {u}}}^T {\mathbf {F}}^{{\mathbf {u}}}\right) ^S \right) \cdot {\dot{{\mathbf {E}}}}({\hat{{\mathbf {u}}}}) \left( \frac{J^{R}}{J^{M}} \right) ^\chi d\varOmega _{t_{n}} \nonumber \\&\quad + \int _{\varOmega _{t_{n}}} \left( \frac{1}{\text {det}{\mathbf {F}}^{{\mathbf {v}}}} {\mathbf {F}}^{{\mathbf {v}}}\frac{\partial {\mathbf {S}}_f}{\partial {\mathbf {E}}^{{\mathbf {u}}{\mathbf {v}}}} \left( {{\mathbf {F}}^{{\mathbf {u}}{\mathbf {v}}}}^T\nabla \delta {\mathbf {u}}\,{\mathbf {F}}^{{\mathbf {v}}}\right) ^S {\mathbf {F}}^{{\mathbf {v}}}\right) \nonumber \\&\quad \cdot {\dot{{\mathbf {E}}}}({\hat{{\mathbf {u}}}}) \left( \frac{J^{R}}{J^{M}} \right) ^\chi d\varOmega _{t_{n}} \nonumber \\&\quad - \int _{\varOmega _{t_{n}}} \left( {\mathbf {S}}_{g} + \frac{1}{\text {det}{\mathbf {F}}^{{\mathbf {v}}}} {\mathbf {F}}^{{\mathbf {v}}}{\mathbf {S}}_{f} {{\mathbf {F}}^{{\mathbf {v}}}}^T \right) \nonumber \\&\quad \cdot {\dot{{\mathbf {E}}}}({\hat{{\mathbf {u}}}}) \chi \left( \frac{J^R}{J^M}\right) ^{\chi } {{\mathbf {F}}^{{\mathbf {u}}}}^{-T} \cdot \nabla \delta {\mathbf {u}}~d\varOmega _{t_{n}} \nonumber \\&\quad + \int _{\varOmega _{t_{n}}} \left( {\mathbf {S}}_{g} + \frac{1}{\text {det}{\mathbf {F}}^{{\mathbf {v}}}} {\mathbf {F}}^{{\mathbf {v}}}{\mathbf {S}}_{f} {{\mathbf {F}}^{{\mathbf {v}}}}^T \right) \nonumber \\&\quad \cdot \left( {\nabla \delta {\mathbf {u}}}^T \nabla {\hat{{\mathbf {u}}}}\right) ^S \left( \frac{J^{R}}{J^{M}} \right) ^\chi d\varOmega _{t_{n}}. \end{aligned}$$

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Biocca, N., Blanco, P.J., Caballero, D. et al. A biologically-inspired mesh optimizer based on pseudo-material remodeling. Comput Mech (2021).

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  • Mesh motion
  • Mechanobiology
  • Growth and remodeling
  • Fiber-reinforced hyperelasticity
  • Fiber recruitment