An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations

The thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite difference (RBF-FD) framework. Then the boundary conditions are reformulated as a Robin boundary condition with smooth coefficients. The same framework is also used to approximate the boundary curve of the diaphragm cross section based on data obtained from a slice of a computed tomography (CT) scan. To solve the PDE we employ the unfitted least-squares RBF-FD method. This makes it easier to handle the geometry of the diaphragm, which is thin and non-convex. We show numerically that our solution converges with high-order towards a finite element solution evaluated on a fine grid. Through this simplified numerical model we also gain an insight into the challenges associated with the diaphragm geometry and the boundary conditions before approaching a more complex three-dimensional model

Oversampled radial basis function methods for solving partial differential equations

Partial differential equations (PDEs) describe complex real-world phenomena such as weather dynamics,  object deformations, financial trading prices, and fluid-structure interaction.  The solutions of PDEs are commonly used to enhance the understanding of these phenomena and also as leverage to make technological improvements to consumer products. In the present thesis, we  develop numerical methods for solving PDEs using computers. The focus is on radial basis  function (RBF) methods that are appreciated for their high-order accuracy and ease of implementation in higher dimensions, but can  sometimes face numerical stability challenges.  To circumvent the stability issues, we use an oversampled approach to discretize PDEs as opposed to the more commonly used collocated approach.  Throughout the thesis, we mainly use the RBF-generated finite difference (RBF-FD) method, but we also use the RBF partition  of unity method (RBF-PUM) and Kansa's global RBF method in one part of the thesis.  The first two methods are local in the sense that the underlying discretization matrices are sparse,  while the third method is global, leading to dense discretization matrices. In Paper I we improve the stability properties of the RBF-FD method through an oversampling approach  when solving an elliptic model problem with derivative-type boundary conditions, and provide a theoretical analysis.  In Paper II we develop an unfitted RBF-FD method and by that simplify the handling of complex  computational domains by relaxing the requirement that  the set of nodes has to conform to the boundary of the domain. We make the first steps toward a simulation of the thoracic diaphragm in Paper III,  where we use an unfitted RBF-FD method to solve a linear elastic PDE and employ data smoothing to leverage high-order convergence of the numerical solution. In Paper IV we explore the stability properties behind the RBF-FD method, Kansa's method, and RBF-PUM  when they are applied to a  linear time-dependent hyperbolic PDE. We find that Kansa's method and RBF-PUM can become stable under sufficient oversampling of the system of equations.  On the other hand, the insufficient regularity of the numerical solution prevents the RBF-FD method from being stable in time, no matter the oversampling.  In Paper V we use the residual viscosity stabilization framework to locally stabilize the Gibbs phenomenon present in the RBF-FD solutions  to shock-inducing nonlinear time-dependent conservation laws such as the compressible Euler system of equations

Oversampled radial basis function methods for solving partial differential equations

Partial differential equations (PDEs) describe complex real-world phenomena such as weather dynamics,  object deformations, financial trading prices, and fluid-structure interaction.  The solutions of PDEs are commonly used to enhance the understanding of these phenomena and also as leverage to make technological improvements to consumer products. In the present thesis, we  develop numerical methods for solving PDEs using computers. The focus is on radial basis  function (RBF) methods that are appreciated for their high-order accuracy and ease of implementation in higher dimensions, but can  sometimes face numerical stability challenges.  To circumvent the stability issues, we use an oversampled approach to discretize PDEs as opposed to the more commonly used collocated approach.  Throughout the thesis, we mainly use the RBF-generated finite difference (RBF-FD) method, but we also use the RBF partition  of unity method (RBF-PUM) and Kansa's global RBF method in one part of the thesis.  The first two methods are local in the sense that the underlying discretization matrices are sparse,  while the third method is global, leading to dense discretization matrices. In Paper I we improve the stability properties of the RBF-FD method through an oversampling approach  when solving an elliptic model problem with derivative-type boundary conditions, and provide a theoretical analysis.  In Paper II we develop an unfitted RBF-FD method and by that simplify the handling of complex  computational domains by relaxing the requirement that  the set of nodes has to conform to the boundary of the domain. We make the first steps toward a simulation of the thoracic diaphragm in Paper III,  where we use an unfitted RBF-FD method to solve a linear elastic PDE and employ data smoothing to leverage high-order convergence of the numerical solution. In Paper IV we explore the stability properties behind the RBF-FD method, Kansa's method, and RBF-PUM  when they are applied to a  linear time-dependent hyperbolic PDE. We find that Kansa's method and RBF-PUM can become stable under sufficient oversampling of the system of equations.  On the other hand, the insufficient regularity of the numerical solution prevents the RBF-FD method from being stable in time, no matter the oversampling.  In Paper V we use the residual viscosity stabilization framework to locally stabilize the Gibbs phenomenon present in the RBF-FD solutions  to shock-inducing nonlinear time-dependent conservation laws such as the compressible Euler system of equations

Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws

In this paper, we solve nonlinear conservation laws using the radial basis function generated finite difference (RBF-FD) method. Nonlinear conservation laws have solutions that entail strong discontinuities and shocks, which give rise to numerical instabilities when the solution    is approximated by a numerical method. We introduce a residual-based artificial viscosity (RV) stabilization framework adjusted to the RBF-FD method, where the residual of the conservation law adaptively locates discontinuities and shocks. The RV stabilization framework is applied to the collocation RBF-FD method and the oversampled RBF-FD method. Computational tests confirm that the stabilized methods are reliable and accurate in solving scalar conservation laws and conservation law systems such as compressible Euler equations

Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws

In this paper, we solve nonlinear conservation laws using the radial basis function generated finite difference (RBF-FD) method. Nonlinear conservation laws have solutions that entail strong discontinuities and shocks, which give rise to numerical instabilities when the solution    is approximated by a numerical method. We introduce a residual-based artificial viscosity (RV) stabilization framework adjusted to the RBF-FD method, where the residual of the conservation law adaptively locates discontinuities and shocks. The RV stabilization framework is applied to the collocation RBF-FD method and the oversampled RBF-FD method. Computational tests confirm that the stabilized methods are reliable and accurate in solving scalar conservation laws and conservation law systems such as compressible Euler equations

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ2-norm intrinsic to each of the three methods.The results show that Kansa's method and RBF-PUM can be ℓ2-stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not ℓ2-stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ2-norm intrinsic to each of the three methods.The results show that Kansa's method and RBF-PUM can be ℓ2-stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not ℓ2-stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations

Geometry Reconstruction from Noisy Data using a Radial Basis Function Partition of Unity Method

International audienceManual three-dimensional segmentation of medical images results in noisy data sets representing three-dimensional objects. Based on this data, we look at how to perform a smooth object reconstruction. In particular, we are interested in the diaphragm, which is a thin curved volume. We use a partition of unity method where local object representations in each patch are blended into a global reconstruction. We use principal component analysis of the local data to align the local approximations with the data. Patches are adaptively refined based on local curvature. Due to the independence of the local approximations, we can increase the resolution in the thin dimension locally in each patch. We use infinitely smooth radial basis functions (RBF) to form a level set function with the object surface as its zero level set. Least squares approximation of the location, gradients, and values outside the object is employed to handle the noise in the data set. We evaluate the resulting reconstruction in terms of residual with respect to the initial data, local curvature, and visual appearance. We present guidelines for how to choose the method parameters, and investigate how they affect the result