# Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Coupled Model of Linear Poroelasticity and Chemotaxis

## 3. Linear Stability Analysis and Dispersion Relation

#### 3.1. Preliminaries

**Proposition 1.**

**Definition 1.**

**Definition 2.**

**Box 1.**Model parameters used in the linear stability analysis of the poro-mechano-chemical system

#### 3.2. Spatially Homogeneous Distributions

#### 3.3. Zero Chemotaxis

#### 3.4. Uncoupled System

**A1**)–(

**B1**)) present the patterning space based on the implicit functions defined in (14a)–(14c) for the $({m}_{0},\alpha )$ and $({\kappa}_{1},{\kappa}_{4})$ space. The conditions are represented in both Figures by a plain-green curve (14a), a red-dot-dashed curve (14b), and a blue-dashed curve (14c). The boundaries present very similar results to the linear analysis from [6]. Here, we accentuate the region by filling it with light grey. Panels (

**A2**)–(

**A3**) and (

**B2**)–(

**B3**) in Figure 1 portray the functions ${a}_{0}\left({k}^{2}\right)$ (parameter condition) and $\lambda \left({k}^{2}\right)$ (dispersion relation) for the studied parameters ${m}_{0}$ and ${\kappa}_{4}$. We present in different colours the behaviour of the system for the critical parameter value (green curve), as well as parameter values obtained by increasing and decreasing by 25% and 50% the critical value, where we recall that fixed parameters are taken from Box 1. We see that moving above or below the critical value results in a finite interval of wave numbers for which the complex eigenvalue presented positive real component and thus leading to instability (patterning). As illustrated also in [6], for a specific parameter set, panels (

**A1**)–(

**A3**) from Figure 1 show that a sufficiently dense dermis cells concentration is necessary to produce instability. While increasing the chemotaxis sensibility of the system we spread the range of possible mesenchymal cells density, by the u-shape form of the boundary, a sufficiently high density of cells can prevent pattern formation (see Figure 1(

**A1**)). The capacity of BMP to deactivate the epithelium by increasing ${\kappa}_{4}$, rapidly decreases the patterning ability of the system.

#### 3.5. Zero Activation/Inactivation of Epithelium

#### 3.6. General Case

`roots`, after constructing the associated vector of polynomial coefficients). In addition, one needs to satisfy some additional conditions leading to ${d}_{0}<0$ for any positive ${k}^{2}$. More specifically, the constraints consist of at least one of the Routh–Hurwitz conditions [34], stated as

**C1**). The dispersion relation for the $({m}_{0},\alpha )$- and $({\kappa}_{1},{\kappa}_{2})$– parameter spaces illustrates how, for the same fixed set of parameters from Box 1, the poromechanical coupling leads to a decrease of the critical parameter value (Figure 2(

**A3,B3**)). Unfortunately, the ease to push the system to instability is counter-balanced by a more restrictive space of parameters values where patterns can be formed (more precisely, compared to Figure 1(

**A1,B1**) and Figure 2(

**A1,B1,C1**)). This can increase the difficulty in tuning parameters to produce specific patterns. Indeed, zooming into the parameter pair $({m}_{0},\alpha )$ on the region where the boundary conditions cross (see Figure 3), exposes how easily the patterns can disappear after a relatively small variation in parameter values.

## 4. Extension to Finite-Strain Poroelasticity and Growth

## 5. Numerical Tests

#### 5.1. Discretisation and Implementation

`FEniCS`[41]. The solution of all nonlinear systems was carried out with Newton’s method using a tolerance of ${10}^{-7}$.on either the relative or absolute ${\ell}^{2}-$norm of the vector residuals, and the linear systems on each step were solved with the distributed direct solver MUMPS. For the 3D case we used the

`Firedrake`library [42] (mainly due to its facility in handling block preconditioners). We provide details regarding the preconditioners in Section 5.3. The nonlinear tolerances are taken identical to the 2D case. For the linear solvers, we use a GMRES with the nested Schur preconditioner recently proposed in [8], with an absolute tolerance of ${10}^{-12}$ and a relative one of ${10}^{-2}$. The use of such a big relative tolerance yields a simplified inexact-Newton method [43] that results in more nonlinear iterations requiring much less time.

#### 5.2. Mesh Independence Study

#### 5.3. Efficient Preconditioners

- Chemotaxis. This problem considers four similar building block physics, consisting essentially of three parabolic problems $(m,f,b)$ and one algebraic constraint $\left(e\right)$. We consider an additive block solver, meaning that we use a block-wise Jacobi preconditioner with only the diagonal block of each variable. At the block level, we consider the action of an AMG preconditioner for the parabolic problems and the action of a Jacobi preconditioner for the algebraic one.
- Poromechanics. The poromechanics block is more difficult, which is reflected in the complexity of the preconditioner under consideration. It is based on the block preconditioner proposed in [8] for large-strain poromechanics, where we use a lower Schur complement block factorisation with the fields $u$ and $(\psi ,p)$. For such a block we consider the action of an AMG preconditioner, whereas for the corresponding Schur complement block we consider instead a sparse representation given by an ad hoc extension of the fixed-stress splitting scheme proposed in [25]. For this, we add two stabilisation terms given by$${\int}_{{\Omega}_{0}}{\beta}_{s}\psi {\psi}^{*}+{\beta}_{f}p{p}^{*}\phantom{\rule{0.166667em}{0ex}}dx,$$

#### 5.4. Suppressed Solid Motion vs. Periodic Boundary Traction

#### 5.5. Finite Growth in 2D

#### 5.6. Finite Growth in 3D

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

## References

- Keller, E.F.; Segel, L.A. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol.
**1970**, 26, 399–415. [Google Scholar] [CrossRef] - Yang, X.; Dormann, D.; Münsterberg, A.E.; Weijer, C.J. Cell movement patterns during gastrulation in the chick are controlled by positive and negative chemotaxis mediated by FGF4 and FGF8. Dev. Cell
**2002**, 3, 425–437. [Google Scholar] [CrossRef] [Green Version] - Lin, C.M.; Jiang, T.X.; Baker, R.E.; Maini, P.K.; Widelitz, R.B.; Chuong, C.M. Spots and stripes: Pleomorphic patterning of stem cells via p-ERK-dependent cell chemotaxis shown by feather morphogenesis and mathematical simulation. Dev. Biol.
**2009**, 334, 369–382. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mou, C.; Pitel, F.; Gourichon, D.; Vignoles, F.; Tzika, A.; Tato, P.; Yu, L.; Burt, D.W.; Bed’Hom, B.; Tixier-Boichard, M.; et al. Cryptic patterning of avian skin confers a developmental facility for loss of neck feathering. PLoS Biol.
**2011**, 9, e1001028. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Painter, K.J.; Hunt, G.S.; Wells, K.L.; Johansson, J.A.; Headon, D.J. Towards an integrated experimental–theoretical approach for assessing the mechanistic basis of hair and feather morphogenesis. Interface Focus
**2012**, 2, 433–450. [Google Scholar] [CrossRef] - Painter, K.J.; Ho, W.; Headon, D.J. A chemotaxis model of feather primordia pattern formation during avian development. J. Theor. Biol.
**2018**, 437, 225–238. [Google Scholar] [CrossRef] [Green Version] - Coussy, O. Poromechanics; John Wiley & Sons: Hoboken, NJ, USA, 2004. [Google Scholar]
- Barnafi, N.; Gómez-Vargas, B.; Lourenço, W.J.; Reis, R.F.; Rocha, B.M.; Lobosco, M.; Ruiz-Baier, R.; Santos, R.W.d. Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema. J. Sci. Comput.
**2022**, 92, e92. [Google Scholar] [CrossRef] - Lourenço, W.d.J.; Reis, R.F.; Ruiz-Baier, R.; Rocha, B.M.; Santos, R.W.d.; Lobosco, M. A poroelastic approach for modelling myocardial oedema in acute myocarditis. Front. Physiol.
**2022**, 13, e888515. [Google Scholar] [CrossRef] [PubMed] - Barnafi, N.; Gregorio, S.D.; Dedè, L.; Zunino, P.; Vergara, C.; Quarteroni, A. A multiscale poromechanics model integrating myocardial perfusion and systemic circulation. SIAM J. Appl. Math.
**2021**, 82, 1113–1660. [Google Scholar] - Vuong, A.-T.; Yoshihara, L.; Wall, W.A. A general approach for modeling interacting flow through porous media under finite deformations. Comput. Methods Appl. Mech. Eng.
**2015**, 283, 1240–1259. [Google Scholar] [CrossRef] - Berger, L.; Bordas, R.; Burrowes, K.; Grau, V.; Tavener, S.; Kay, D. A poroelastic model coupled to a fluid network with applications in lung modelling. Int. J. Numer. Methods Biomed. Eng.
**2016**, 32, e02731. [Google Scholar] [CrossRef] [PubMed] - Vilaca, L.M.D.O.; Gómez-Vargas, B.; Kumar, S.; Ruiz-Baier, R.; Verma, N. Stability analysis for a new model of multi-species convection-diffusion-reaction in poroelastic tissue. Appl. Math. Model.
**2020**, 84, 425–446. [Google Scholar] [CrossRef] - Moeendarbary, E.; Valon, L.; Fritzsche, M.; Harris, A.R.; Moulding, D.A.; Thrasher, A.J.; Stride, E.; Mahadevan, L.; Charras, G.T. The cytoplasm of living cells behaves as a poroelastic material. Nat. Mater.
**2013**, 12, e3517. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Collis, J.; Brown, D.L.; Hubbard, M.E.; O’Dea, R.D. Effective equations governing an active poroelastic medium. Proc. R. Soc. A
**2017**, 473, e20160755. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Penta, R.; Ambrosi, D.; Shipley, R.J. Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math.
**2014**, 67, 69–91. [Google Scholar] [CrossRef] [Green Version] - Jones, G.W.; Chapman, S.J. Modeling growth in biological materials. SIAM Rev.
**2012**, 54, 52–118. [Google Scholar] [CrossRef] - Kuhl, E. Growing matter: A review of growth in living systems. J. Mech. Behav. Biomed. Mater.
**2014**, 29, 529–543. [Google Scholar] [CrossRef] - Mascheroni, P.; Carfagna, M.; Grillo, A.; Boso, D.P.; Schrefler, B.A. An avascular tumor growth model based on porous media mechanics and evolving natural states. Math. Mech. Solids
**2018**, 23, 686–712. [Google Scholar] [CrossRef] - Moreo, P.; Gaffney, E.A.; García-Aznar, J.M.; Doblaré, M. On the modelling of biological 795 patterns with mechanochemical models: Insights from analysis and computation. Bull. Math. Biol.
**2010**, 72, 400–431. [Google Scholar] [CrossRef] - Murray, J.D.; Maini, P.K.; Tranquillo, R.T. Mechanochemical models for generating biological pattern and form in development. Phys. Rep.
**1988**, 171, 59–84. [Google Scholar] [CrossRef] [Green Version] - Radszuweit, M.; Engel, H.; Bär, M. An active poroelastic model for mechanochemical patterns in protoplasmic droplets of physarum polycephalum. PLoS ONE
**2014**, 9, e99220. [Google Scholar] [CrossRef] [PubMed] - Barnafi, N.; Zunino, P.; Dedè, L.; Quarteroni, A. Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model. Comput. Math. Appl.
**2021**, 91, 202–228. [Google Scholar] [CrossRef] - Berger, L.; Bordas, R.; Kay, D.; Tavener, S. A stabilized finite element method for finite-strain three-field poroelasticity. Comput. Mech.
**2017**, 60, 51–68. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Borregales, M.; Radu, F.A.; Kumar, K.; Nordbotten, J.M. Robust iterative schemes for non-linear poromechanics. Computat. Geosci.
**2018**, 22, 1021–1038. [Google Scholar] [CrossRef] [Green Version] - Costanzo, F.; Miller, S.T. An arbitrary Lagrangian-Eulerian finite element formulation for a poroelasticity problem stemming from mixture theory. Comput. Methods Appl. Mech. Eng.
**2017**, 323, 64–97. [Google Scholar] [CrossRef] - Korsawe, J.; Starke, G.; Wang, W.; Kolditz, O. Finite element analysis of poro-elastic consolidation in porous media: Standard and mixed approaches. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 1096–1115. [Google Scholar] [CrossRef] - Verma, N.; Gómez-Vargas, B.; Vilaca, L.M.D.O.; Kumar, S.; Ruiz-Baier, R. Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media. Appl. Anal.
**2022**, 101, 4914–4941. [Google Scholar] [CrossRef] - Kadeethum, T.; Lee, S.; Ballarind, F.; Choo, J.; Nick, H.M. A locally conservative mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. Comput. Geosci.
**2021**, 152, e104774. [Google Scholar] [CrossRef] - Armstrong, M.H.; Tepole, A.B.; Kuhl, E.; Simon, B.R.; Geest, J.P.V. A finite element model for mixed porohyperelasticity with transport, swelling, and growth. PLoS ONE
**2016**, 11, e0152806. [Google Scholar] [CrossRef] [Green Version] - Jin, L.; Zoback, M.D. Fully dynamic spontaneous rupture due to quasi-static pore pressure and poroelastic effects: An implicit nonlinear computational model of fluid-induced seismic events. J. Geophys. Res. Solid Earth
**2018**, 123, 9430–9468. [Google Scholar] [CrossRef] [Green Version] - Luo, P.; Rodrigo, C.; Gaspar, F.J.; Oosterlee, C.W. Multigrid method for nonlinear poroelasticity equations. Comput. Visual. Sci.
**2015**, 17, 255–265. [Google Scholar] [CrossRef] - Showalter, R.E. Diffusion in poro-elastic media. J. Math. Anal. Appl.
**2000**, 251, 310–340. [Google Scholar] [CrossRef] [Green Version] - Routh, E.J. A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion; Macmillan and Company: London, UK, 1877. [Google Scholar]
- Lee, E.H. Elastic-plastic deformation at finite strains. J. Appl. Mech.
**1969**, 36, 1–6. [Google Scholar] [CrossRef] - Rodriguez, E.K.; Hoger, A.; McCulloch, A.D. Stress-dependent finite growth in soft elastic tissues. J. Biomech.
**1994**, 27, 455–467. [Google Scholar] [CrossRef] - Kida, N.; Morishita, Y. Continuum mechanical modeling of developing epithelial tissues with anisotropic surface growth. Finite Elem. Anal. Des.
**2018**, 144, 49–60. [Google Scholar] [CrossRef] - Amar, M.B.; Goriely, A. Growth and instability in elastic tissues. J. Mech. Phys. Solids
**2005**, 53, 2284–2319. [Google Scholar] [CrossRef] - Giverso, C.; Scianna, M.; Grillo, A. Growing avascular tumours as elastoplastic bodies by the theory of evolving natural configurations. Mech. Res. Commun.
**2015**, 68, 31–39. [Google Scholar] [CrossRef] - Braess, D.; Ming, P. A finite element method for nearly incompressible elasticity problems. Math. Comp.
**2005**, 74, 25–52. [Google Scholar] [CrossRef] [Green Version] - Alnæs, M.S.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M.E.; Wells, G.N. The FEniCS project version 1.5. Arch. Numer. Softw.
**2015**, 3, 9–23. [Google Scholar] - Rathgeber, F.; Ham, D.A.; Mitchell, L.; Lange, M.; Luporini, F.; McRae, A.T.T.; Bercea, G.T.; Markall, G.R.; Kelly, P.H.J. Firedrake: Automating the finite element method by composing abstractions. ACM T. Math. Softw. (TOMS)
**2016**, 43, 1–27. [Google Scholar] [CrossRef] [Green Version] - Dembo, R.S.; Eisenstat, S.C.; Steihaug, T. Inexact Newton methods. SIAM J. Numer. Anal.
**1982**, 19, 400–408. [Google Scholar] [CrossRef] - Dervaux, J.; Ciarletta, P.; Amar, M.B. Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Föppl–von Kármaán limit. J. Mech. Phys. Solids
**2009**, 57, 458–471. [Google Scholar] [CrossRef] - Vilaca, L.M.D.O.; Milinkovitch, M.C.; Ruiz-Baier, R. Numerical approximation of a 3D mechanochemical interface model for skin patterning. J. Comput. Phys.
**2019**, 384, 283–404. [Google Scholar] - Bociu, L.; Guidoboni, G.; Sacco, R.; Webster, J.T. Analysis of nonlinear poro-elastic and poro-visco-elastic models. Arch. Ration. Mech. Anal.
**2016**, 222, 1445–1519. [Google Scholar] [CrossRef] [Green Version] - Kaouri, K.; Méndez, P.E.; Ruiz-Baier, R. Mechanochemical models for calcium waves in embryonic epithelia. Vietnam J. Math.
**2022**, 50, 947–975. [Google Scholar] [CrossRef]

**Figure 1.**Patterning space, parameter condition and dispersion relations for the uncoupled poro–mechano–chemical model. (

**A1**) Predicted patterning space for a selected interval in $({m}_{0},\alpha )$ parameter space: (green plain) boundary constructed from (14a); (red-dashed) boundary coming from (14b); (blue–dot–dashed) boundary built from (14c). (

**A2**) Parameter coefficient condition a

_{0}Curves are drawn from the critical value ${m}_{0}^{c}$ (green) and for 25% and 50% increase/decrease parameter values. (

**A3**) Associated dispersion relations. Colour code is kept identical with (

**A2**). (

**B1**–

**B3**) Similar analysis for the $({\kappa}_{1},{\kappa}_{4})$ parameter space.

**Figure 2.**Patterning space, parameter condition and dispersion relations for the coupled poro–mechano–chemical model. (

**A1**): predicted patterning space for a selected interval in $({m}_{0},\alpha )$. Parameter space: (black plain) boundary built from (15a) for ${\theta}_{1}$; (green-dashed) boundary built from (15a) for ${\theta}_{2}$; (red-dot-dot) boundary built from (15b); (blue–dot–dashed) boundary from (15c); (magenta-dot-plain) boundary built from (16). (

**A2**): Coefficient condition on ${d}_{0}$. Curves are drawn from the critical value ${m}_{0}^{c}$ (green) and for the 25% and 50% increase/decrease parameter value. (

**A3**): associated dispersion relations. Colour code is kept identical with (

**A2**). (

**B1**–

**B3**): similar analysis for the $({\kappa}_{1},{\kappa}_{4})$ parameter space. (

**C1**–

**C3**): similar analysis for the $(\tau ,{\xi}_{f})$ parameter space.

**Figure 3.**Patterning space for the coupled poro-mechano-chemical model. Selected interval in the $({m}_{0},\alpha )$ parameter space. (black plain) boundary built from (15a) for ${\theta}_{1}$; (green-dashed) boundary from (15a) for ${\theta}_{2}$; (red-dot-dot) boundary from (15b); (blue-dot-dashed) boundary from (15c); (magenta-dot-plain) boundary built from (16).

**Figure 4.**Decomposition of the tensor gradient of deformation $\mathbf{F}$ into pure growth ${\mathbf{F}}_{g}$ and an elastic deformation tensor ${\mathbf{F}}_{e}$. The intermediate configuration $\tilde{\Omega}$, between the undeformed reference state ${\Omega}_{0}$ and the current/final configuration $\Omega $ including growth and elastic response with stress, is an incompatible and stress-free growth state.

**Figure 5.**Evolution of the mesenchymal cell concentration at $t=160,280,400,520$, under no deformation (

**top**panels); and epithelium, FGF, and BMP concentrations at $t=520$ (

**bottom**).

**Figure 6.**Evolution of the mesenchymal cell concentration at $t=160,280,400,520$ under periodic traction applied on the top edge of the domain (

**top**); and snapshots of fluid pressure, epithelium, FGF, and BMP at $t=520$ (

**bottom**row).

**Figure 7.**Evolution of the mesenchymal cell concentration under finite growth using the formulation (30), snapshots are taken at $t=160,280,400,520$ (first row). Bottom: plots at $t=520$ of fluid pressure, solid pressure, epithelium, FGF, and BMP. Mechano-chemical coupling governed by ${\xi}_{f}=0.5$.

**Figure 8.**Evolution of the mesenchymal cell concentration under finite growth using the formulation (30), snapshots are taken at $t=160,280,400,520$ (top). Second row: plots at $t=520$ of fluid pressure, solid pressure, epithelium, FGF, and BMP. Simulations using ${\xi}_{f}=-0.5$.

**Figure 9.**Evolution of the mesenchymal cell concentration under finite growth using the formulation (29), snapshots are taken at $t=160,280,400,520$ (top). We also display the fluid and solid pressures (second row), the mesenchymal and epithelium concentrations (third row), and the FGF and BMP (fourth row).

**Table 1.**Accuracy verification test using continuous and piecewise linear elements for the approximation of all variables. Error history (number of degrees of freedom, errors in the ${H}^{1}$-norm, and experimental convergence rates) with respect to smooth manufactured solutions computed at $t=10$. The symbol * indicates that the convergence rate is not computed for the coarsest level.

DoF | $\parallel \mathit{u}-{\mathit{u}}_{\mathit{h}}{\parallel}_{1}$ | r | $\parallel \mathit{p}-{\mathit{p}}_{\mathit{h}}{\parallel}_{1}$ | r | $\parallel \mathit{m}-{\mathit{m}}_{\mathit{h}}{\parallel}_{1}$ | r | $\parallel \mathit{e}-{\mathit{e}}_{\mathit{h}}{\parallel}_{1}$ | r | $\parallel \mathit{f}-{\mathit{f}}_{\mathit{h}}{\parallel}_{1}$ | r | $\parallel \mathit{b}-{\mathit{b}}_{\mathit{h}}{\parallel}_{1}$ | r |
---|---|---|---|---|---|---|---|---|---|---|---|---|

64 | $1.0\times {10}^{2}$ | * | $1.4\times {10}^{3}$ | * | $1.0\times {10}^{2}$ | * | $8.0\times {10}^{0}$ | * | $2.8\times {10}^{1}$ | * | $2.9\times {10}^{1}$ | * |

176 | $3.0\times {10}^{1}$ | 1.77 | $6.4\times {10}^{2}$ | 1.16 | $2.8\times {10}^{0}$ | 1.88 | $4.9\times {10}^{0}$ | 0.70 | $1.9\times {10}^{1}$ | 0.67 | $1.8\times {10}^{1}$ | 0.63 |

568 | $1.1\times {10}^{1}$ | 1.19 | $3.6\times {10}^{2}$ | 0.97 | $1.1\times {10}^{0}$ | 1.36 | $4.1\times {10}^{0}$ | 0.57 | $1.4\times {10}^{1}$ | 0.53 | $1.4\times {10}^{1}$ | 0.56 |

2024 | $5.0\times {10}^{0}$ | 1.08 | $1.7\times {10}^{2}$ | 1.19 | $5.1\times {10}^{-1}$ | 1.09 | $3.3\times {10}^{0}$ | 0.61 | $8.5\times {10}^{0}$ | 0.73 | $8.6\times {10}^{0}$ | 0.73 |

7624 | $2.4\times {10}^{0}$ | 1.02 | $8.3\times {10}^{1}$ | 1.11 | $2.5\times {10}^{-1}$ | 1.02 | $2.2\times {10}^{0}$ | 0.76 | $4.4\times {10}^{0}$ | 0.96 | $4.4\times {10}^{0}$ | 0.96 |

29,576 | $1.3\times {10}^{0}$ | 0.98 | $4.3\times {10}^{1}$ | 0.94 | $1.3\times {10}^{-1}$ | 1.01 | $1.3\times {10}^{0}$ | 0.82 | $2.2\times {10}^{0}$ | 1.00 | $2.2\times {10}^{0}$ | 0.99 |

116,488 | $6.6\times {10}^{-1}$ | 0.98 | $2.4\times {10}^{1}$ | 0.92 | $6.3\times {10}^{-2}$ | 1.00 | $6.5\times {10}^{-1}$ | 0.95 | $1.1\times {10}^{0}$ | 1.00 | $1.1\times {10}^{0}$ | 1.00 |

**Table 2.**Mesh independence test. Approximate number of triangular elements and of degrees of freedom, relative computational time, and relative error with respect to a fine mesh computation.

Mesh | Cardinality | Total DoFs | CPU Time | Error |
---|---|---|---|---|

I | 30 K | 110 K | 15% | 9% |

II | 60 K | 218 K | 25% | 5% |

III | 120 K | 430 K | 40% | 4% |

**Table 3.**Performance of the nonlinear solver used with the preconditioner detailed in Section 5.3. We show the total CPU time on the nonlinear solver on a single timestep, together with the maximum number of GMRES iterations in parenthesis.

DoFs | 1 CPU | 2 CPUs | 4 CPUs | 8 CPUs | 12 CPUs | 16 CPUs |
---|---|---|---|---|---|---|

3528 | 0.57 (10) | 0.56 (10) | 0.68 (10) | 0.72 (10) | 0.77 (10) | 0.7 (10) |

20,172 | 1.01 (10) | 1.00 (9) | 1.01 (10) | 0.93 (10) | 1.25 (10) | 0.93 (10) |

59,536 | 2.35 (11) | 1.95 (9) | 1.46 (10) | 1.31 (9) | 1.03 (10) | 1.29 (10) |

131,220 | 5.43 (11) | 3.44 (10) | 2.48 (10) | 1.81 (10) | 1.63 (10) | 1.6 (10) |

244,824 | 10.72 (11) | 6.36 (11) | 4.35 (11) | 2.7 (10) | 2.59 (10) | 2.34 (10) |

409,948 | 19.03 (12) | 12.93 (12) | 7.09 (12) | 4.7 (12) | 3.72 (12) | 3.26 (12) |

636,192 | 31.57 (12) | 20.14 (12) | 11.27 (12) | 7.11 (12) | 5.38 (12) | 4.72 (12) |

933,156 | 49.81 (12) | 29.2 (12) | 16.24 (12) | 9.73 (12) | 7.55 (12) | 6.82 (12) |

(a) Performance of chemotaxis solver. | ||||||

DoFs | 1 CPU | 2 CPUs | 4 CPUs | 8 CPUs | 12 CPUs | 16 CPUs |

16,893 | 2.5 (8) | 1.8 (8) | 1.8 (8) | 2.05 (8) | 1.91 (8) | 2.17 (8) |

108,501 | 12.13 (12) | 7.27 (12) | 4.8 (12) | 3.11 (12) | 2.8 (12) | 2.89 (12) |

337,229 | 39.71 (13) | 22.62 (14) | 12.31 (14) | 7.57 (14) | 6.6 (14) | 5.72 (14) |

765,477 | 98.6 (14) | 52.06 (14) | 28.45 (14) | 16.5 (14) | 12.57 (14) | 10.79 (14) |

1,455,645 | 202.15 (14) | 102.76 (14) | 56.51 (14) | 32.32 (14) | 23.83 (14) | 19.58 (14) |

2,470,133 | 350.59 (14) | 192.52 (14) | 99.84 (14) | 55.04 (14) | 40.24 (14) | 33.75 (14) |

3,871,341 | 598.98 (15) | 324.14 (15) | 168.48 (15) | 89.02 (15) | 63.27 (15) | 54.29 (15) |

5,721,669 | 869.06 (15) | 462.3 (15) | 242.9 (15) | 130.09 (15) | 98.15 (15) | 78.02 (15) |

(b) Performance of poromechanics solver. |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Barnafi, N.A.; De Oliveira Vilaca, L.M.; Milinkovitch, M.C.; Ruiz-Baier, R.
Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning. *Mathematics* **2022**, *10*, 4096.
https://doi.org/10.3390/math10214096

**AMA Style**

Barnafi NA, De Oliveira Vilaca LM, Milinkovitch MC, Ruiz-Baier R.
Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning. *Mathematics*. 2022; 10(21):4096.
https://doi.org/10.3390/math10214096

**Chicago/Turabian Style**

Barnafi, Nicolás A., Luis Miguel De Oliveira Vilaca, Michel C. Milinkovitch, and Ricardo Ruiz-Baier.
2022. "Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning" *Mathematics* 10, no. 21: 4096.
https://doi.org/10.3390/math10214096