Derivation and simulation of a two-phase fluid deformable surface model

We consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modeled by incompressible surface Navier-Stokes-Cahn-Hilliard-like equations with bending forces. We derive this model using the Lagrange-D'Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor-Hood element for the surface Navier-Stokes part, a semi-implicit approach for the Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified sub-problems

The interplay of geometry and coarsening in multicomponent lipid vesicles under the influence of hydrodynamics

We consider the impact of surface hydrodynamics on the interplay between curvature and composition in coarsening processes on model systems for biomembranes. This includes scaling laws and equilibrium configurations, which are investigated by computational studies of a surface two-phase flow problem with additional phase-depending bending terms. These additional terms geometrically favor specific configurations. We find that as in 2D the effect of hydrodynamics strongly depends on the composition. In situations where the composition allows a realization of a geometrically favored configuration, the hydrodynamics enhances the evolution into this configuration. We restrict our model and numerics to stationary surfaces and validate the numerical approach with various benchmark problems and convergence studies

Numerical methods for Shallow Water Equations on regular surfaces

Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this thesis we derive an intrinsic shallow water model starting from the Navier-Stokes equations defined on a local reference frame anchored on the bottom surface. The resulting equations are characterized by non-autonomous flux functions and source terms embodying only geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier-Stokes equations with respect to a geometry-based order parameter. We then derive numerical discretization schemes compatibles with the intrinsic setting of the formulation, starting from studying a first order upwind Godunov Finite Volume scheme intrinsically defined on the bottom surface. We analyze convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures. The low-order discretization method is subsequently extended to the Discontinuous Galerkin framework. We implement a linear version of the DG scheme defined intrinsically on the surface and we start from the resolution of the scalar transport equation. We test the scheme for convergence and then we move towards the intrinsic shallow water model. Simulations on synthetic test cases are reported and the improvement with respect to the first order finite volume discretization is clearly visible. Finally, we consider a finite element method for advection-diffusion-reaction equations on surfaces. Unlike many previous techniques, this approach is based on the geometrically intrinsic formulation and the resulting finite element method is fully intrinsic to the surface. In the last part of this work, we lay out in detail the formulation and compare it to a well-established finite element scheme for surface PDEs. We then evaluate the method for several steady and transient problems involving both diffusion and advection-dominated regime. The experimental results show the theoretically expected convergence rates and good performance of the established finite element methods

Numerical methods for Shallow Water Equations on regular surfaces

Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this thesis we derive an intrinsic shallow water model starting from the Navier-Stokes equations defined on a local reference frame anchored on the bottom surface. The resulting equations are characterized by non-autonomous flux functions and source terms embodying only geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier-Stokes equations with respect to a geometry-based order parameter. We then derive numerical discretization schemes compatibles with the intrinsic setting of the formulation, starting from studying a first order upwind Godunov Finite Volume scheme intrinsically defined on the bottom surface. We analyze convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures. The low-order discretization method is subsequently extended to the Discontinuous Galerkin framework. We implement a linear version of the DG scheme defined intrinsically on the surface and we start from the resolution of the scalar transport equation. We test the scheme for convergence and then we move towards the intrinsic shallow water model. Simulations on synthetic test cases are reported and the improvement with respect to the first order finite volume discretization is clearly visible. Finally, we consider a finite element method for advection-diffusion-reaction equations on surfaces. Unlike many previous techniques, this approach is based on the geometrically intrinsic formulation and the resulting finite element method is fully intrinsic to the surface. In the last part of this work, we lay out in detail the formulation and compare it to a well-established finite element scheme for surface PDEs. We then evaluate the method for several steady and transient problems involving both diffusion and advection-dominated regime. The experimental results show the theoretically expected convergence rates and good performance of the established finite element methods

Numerical methods for Shallow Water Equations on regular surfaces

Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this thesis we derive an intrinsic shallow water model starting from the Navier-Stokes equations defined on a local reference frame anchored on the bottom surface. The resulting equations are characterized by non-autonomous flux functions and source terms embodying only geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier-Stokes equations with respect to a geometry-based order parameter. We then derive numerical discretization schemes compatibles with the intrinsic setting of the formulation, starting from studying a first order upwind Godunov Finite Volume scheme intrinsically defined on the bottom surface. We analyze convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures. The low-order discretization method is subsequently extended to the Discontinuous Galerkin framework. We implement a linear version of the DG scheme defined intrinsically on the surface and we start from the resolution of the scalar transport equation. We test the scheme for convergence and then we move towards the intrinsic shallow water model. Simulations on synthetic test cases are reported and the improvement with respect to the first order finite volume discretization is clearly visible. Finally, we consider a finite element method for advection-diffusion-reaction equations on surfaces. Unlike many previous techniques, this approach is based on the geometrically intrinsic formulation and the resulting finite element method is fully intrinsic to the surface. In the last part of this work, we lay out in detail the formulation and compare it to a well-established finite element scheme for surface PDEs. We then evaluate the method for several steady and transient problems involving both diffusion and advection-dominated regime. The experimental results show the theoretically expected convergence rates and good performance of the established finite element methods

Geometrically Intrinsic Modeling of Shallow Water Flows

Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this paper we derive an intrinsic shallow water model from the Navier–Stokes equations defined on a local reference frame anchored on the bottom surface. The equations resulting are characterized by non-autonomous flux functions and source terms embodying only the geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier–Stokes equations with respect to a geometry-based order parameter. We then derive a numerical discretization by means of a first order upwind Godunov finite volume scheme intrinsically defined on the bottom surface. We study convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures

Diffusion of tangential tensor fields: numerical issues and influence of geometric properties

We study the diffusion of tangential tensor-valued data on curved surfaces. For this several finite element based numerical methods are collected and used to solve a tangential surface n-tensor heat flow problem. These methods differ with respect to the surface representation used, the required geometric information and the treatment of the tangentiality condition. We highlight the importance of geometric properties and their increasing influence if the tensorial degree changes from n=0 to n>=1. A specific example is presented that illustrates how curvature drastically affects the behavior of the solution.Comment: 28 pages, 6 figure