In this article we propose models and a numerical method for pattern formation on evolving curved surfaces. We formulate reaction-diffusion equations on evolving surfaces using the material transport formula, surface gradients and diffusive conservation laws. The evolution of the surface is defined by a material surface velocity. The numerical method is based on the evolving surface finite element method. The key idea is based on the approximation of Γ by a triangulated surface Γ h consisting of a union of triangles with vertices on Γ. A finite element space of functions is then defined by taking the continuous functions on Γ h which are linear affine on each simplex of the polygonal surface. To demonstrate the capability, flexibility, versatility and generality of our methodology we present results for uniform isotropic growth as well as anisotropic growth of the evolution surfaces and growth coupled to the solution of the reaction-diffusion system. The surface finite element method provides a robust numerical method for solving partial differential systems on continuously evolving domains and surfaces with numerous applications in developmental biology, tumour growth and cell movement and deformation.
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机译：在本文中，我们提出了用于在弯曲曲面上形成图案的模型和数值方法。我们使用物质传输公式，表面梯度和扩散守恒律在不断发展的表面上拟定反应扩散方程。表面的演变由材料表面速度定义。数值方法是基于演化表面有限元法的。关键思想是基于三角表面Γ h 对Γ的近似，该三角形表面由Γ上具有顶点的三角形的并集组成。然后，通过在Γ h 上取连续函数来定义函数的有限元素空间，这些连续函数在多边形表面的每个单形上都是线性仿射。为了证明我们方法的能力，灵活性，多功能性和通用性，我们给出了均匀各向同性生长以及演化表面各向异性生长以及与反应扩散系统解耦合的生长的结果。表面有限元方法为解决连续演化的域和表面上的偏微分系统提供了鲁棒的数值方法，在发展生物学，肿瘤生长以及细胞运动和变形中具有许多应用。
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