Numerical Simulations Of The Artificial Graft Design Problem

In recent years, stenosed artery is one of the most common diseases people faced. One way to deal with the stenosed artery is to transplant a artificial graft on it so that the blood can go through the artificial graft which is called artificial graft design problem. Artificial graft design problem is one of the shape opti-mization problems, which describes a class of constrained optimization problems, which are the objective function minimization process within the constraints of boundary value problems for partial differential equations, the control variable is the geometry of the region. In the artificial graft design problem, there are many difficulty in the numerical simulation. First of all, the solution of the problem should be smooth enough since any error on boundary will make a big error in the solution inside the tube. Secondly, since the computational domain is changing in the optimization process, the mesh should be changed. We need a appropriate mesh update algorithm. Thus, we adopt control points method based on cubic spline interpolation to control the shape of the artificial graft. Non-uniform mesh and moving mesh method is used to improve the efficiency of our algorithm. Cu-bic spline level set method and radial basis function method is also adopted to solve the artificial problem. The results of our work is shown as blow:1. We give a cubic spline algorithm to solve the artificial graft design problem. We adopt a control points method which is based on cubic spline interpo-lation to control the shape of the artificial graft. Thus, the shape of the artificial graft can change more flexibly. Besides, the smoothness of the graft is guaranteed. While the shape of the artificial graft changes, gradi-ent algorithm is used to control the locations of the control points. In the meantime, the initial value of the gradient algorithm is optimized. The op-timal value of the less control points is chosen as the initial value while the number of the control points is more. Non-uniform Delauney triangulation is adopted so that the flow speed is more accurate near the graft wall. Since the artificial graft’shape is always changing, the mesh need to be updated too. In our scheme, moving mesh method is used to update the mesh to avoid large computational cost by re-meshing. In the numerical results, we compare the efficiency of our algorithm and existed algorithms. It shows that our algorithm is more efficient and the value of the objective function of the optimal graft is smaller. In the same time, the effect using different number of control points in the gradient algorithm scheme show that while the number of the control points increase the value of the objective func-tion in the optimal shape decreases. It means it is necessary to increase the number of control points. The algorithm of choosing the optimal value of less control points as the initial value in more control points scheme can save much time especially when the number of the control points is large. In addition, the effect of non-uniform mesh and uniform mesh, re-meshing and moving mesh method are show. We find that the non-uniform mesh is better than the unform one and the computational cost decreases a lot using moving mesh method instead of re-meshing.2. We give a cubic spline level set method in the artificial graft problem. In the shape optimization problem, level set methods is more and more popular. One limitation of level set methods is that it often generate non-smooth interface which affect the solution of the artificial graft design a lot. So we adopt a cubic spline level set algorithm to reconstruct the interface. Based on the traditional level set method, while we get the updated value of the level set function by solving the Hamilton-Jacobi equation. We use cubic spline interpolation method to reconstruct the interface of the artificial graft. The advantage of level set methods is that the mesh is fixed and the state equation and adjoint equation is solved in the whole domain. In numerical results we can see that the optimal artificial graft is smooth and the computational cost using cubic spline level set algorithm is much less than others.3. We adopt a radial basis function(RBF) level set method to solve the ar-tificial graft design problem. RBF level set method aim to parameterize the level set function using RBF. After that the level set function is a series of linear combination of smooth functions. Thus the level set func-tion is smooth and the interface of the artificial graft is smooth too. The Hamilton-Jacobi equation is converted to an ordinary differential equation-s. In the iterative process, the variables are the coefficients of the RBF. And the re-initialization process which is necessary in traditional level set methods can be avoided since the level set function under the RBF scheme will not be too flat or too steep. In the numerical results, the optimal shape of the artificial graft using the RBF level set methods is shown. The effect of it compared with cubic spline level set algorithm is also shown. We find both of them can convergent to optimal shape and it is independent with the initial value of the level set functions. The optimal artificial graft is almost the same using these two algorithms and the computational cost in the cubic spline level set algorithm is less than than the RBF level set algorithm.The paper is organized as follows:In the first chapter, we introduce the background of the artificial graft design problem and the development of optimal shape design problems.In the second chapter, the weak form of Navier-Stokes equations is given. The discrete system applied MINI finite elements and the basis functions on triangle are shown. The assembly of stiff matrix in the numerical algorithm is given. And a brief introduce in Gauss quadrature is shown in the end.In chapter 3, we propose a cubic spline approach algorithm to solve the ar-tificial graft design problem. We adopt a control point method which is based on cubic spline interpolation. The initial value of the gradient algorithm is opti-mized by using the optimal value of the less control points. Non-uniform mesh and moving mesh method is adopted to improve the efficiency of the algorithm. The numerical results show the effect and the high efficiency of our algorithm.In chapter 4, we give the cubic spline level set algorithm and radial basis function level set algorithm to solve the artificial graft design problem. The traditional level set method is introduced firstly. The sensitivity analysis and the derivation of the adjoint equation is then shown. Since the interface generated by the traditional level set methods is not smooth in many cases. We adopt the cubic spline level set algorithm to smooth the interface. In the same time, radial basis function level set algorithm is adopted to parameterize the level set function to ensure the level set function be smooth enough. Numerical results show the effect and efficiency of these algorithms.The last chapter is the conclusion and the outline of the future research work.