Stephen R. Kennon

Generation of solution-adaptive computational grids using optimization

Abstract A new method for generating solution-adaptive computational grids is presented that builds on the requirement that the adapted grid retains maximum possible smoothness and local orthogonality. The approach taken is one of nonlinear optimization, where an objective function combining measures of grid smoothness, local orthogonality, and cell volume control is minimized using a fast iterative scheme. The method is multidimensional by construction, and accepts any arbitrary grid as input, even a grid that is initially overlapped. Several applications to published test problems allow comparisons with some of the existing adaptive grid generation methods. An example of dynamic adaptation to the solution of finite difference equations is also given that demonstrates the error reduction capabilities of the new adaptive grid generation and optimization method, and suggests further applications to practical engineering problems.

OPTIMIZATION OF THREE-DIMENSIONAL COMPUTATIONAL GRIDS

A method for generating and optimizing arbitrary three-dimensional boundary-confo rming computational grids has been developed. The smoothness and local orthogonality of the grid are maximized using a fast iterative procedure, and provision is made for clustering the optimized grid in selected regions. An opimal grid can be obtained iteratively, irrespective of the method used to generate the initial grid. Unacceptable grids and even singular grids (i.e., grids containing regions of overlap) can be made useful for computation using this method. Application of the method to several test cases shows that grids containing regions of overlap are typically untangled in 2-5 iterations and that the conjugate gradient optimization procedure converges to an optimized grid within 25 iterations. Taking advantage of the original properties of this method, a new concept for generating optimal three-dimensional computational grids is proposed. It consists in optimizing a first guess of the desired grid, using an imperfect grid generated by a simple, inexpensive method as input.

Optimum acceleration factors for iterative solution of linear and nonlinear differential systems

Abstract A new acceleration concept for iterative schemes is described. The concept is based on elementary variational calculus, and can be readily implemented in the iterative solution of a wide variety of linear and nonlinear differential systems. The method is not limited to finite difference, finite element or finite volume discretization schemes, but only to schemes that are inherently iterative. Most importantly, the method is exact in the sense that optimal relaxation/acceleration factors can be analytically determined for a class of commonly encountered systems possessing simple nonlinearity. For systems exhibiting complex nonlinearity, the method can be applied in a semi-exact but highly accurate fashion using truncated Taylor series. Without any modification, this acceleration method can be directly applied to existing iterative schemes using either orthogonal or completely arbitrary non-orthogonal computational grids, since the formulation of the method is dependent only on the governing differential system. The described method belongs to the general class of minimal residual techniques, but can be applied to nonlinear systems.

Generalized nonlinear minimal residual (GNLMR) method for iterative algorithms

Abstract Most iterative methods for solving steady-state problems can be shown to be equivalent to solving time-dependent problems of either parabolic or hyperbolic type. The relaxation factor used in accelerating an iterative method to obtain the converged solution plays the same role as the time step size used in advancing the transient solution to the steady state solution for a time-dependent problem. With this transformation, one can expose the mechanism of the acceleration schemes. In the presented study, this time-dependent approach together with the single-iteration, multi-step algorithm are applied to generalize the nonlinear minimal residual (NLMR) method for iterative solutions of linear and nonlinear problems. Most importantly, both theoretical studies and numerical experiments confirm the monotone convergence behavior of the generalized NLMR method. With the multi-step algorithm, it is found that both the rate and the smoothness of convergence of the NLMR method can be improved even further. Several interesting problems that originated from this method are also discussed.

Grid orthogonalization for curvilinear alternating-direction techniques

Abstract A method is developed for an a posteriori iterative improvement to an arbitrary computational grid. Local corrections to the coordinates of the grid points are used to form a global cost function which is minimized with respect to a single parameter. The local corrections and cost function can be constructed to maximize the local smoothness and/or the local orthogonality of the grid. The advantage of this method is that it allows the user to generate an initial grid using any inexpensive method, and then the grid can be improved with respect to both orthogonality and smoothness. This technique was used to generate grids for a finite element alternating-direction method which uses curved elements. A sample transient diffusion problem was solved on a series of grids to investigate the sensitivity of the curvilinear alternating-direction method to grid orthogonalization. The initial grid was highly nonorthogonal and each grid produced by the automatic grid generation program was smoother and more orthogonal. This work shows that the adaptive grid program can be easily used to generate nearly orthogonal grids and it shows that the curvilinear alternating-direction technique is not highly sensitive to nonorthogonality of the grid. It is shown that as long as a grid is somewhat reasonable, the alternating-direction method will perform quite well.

Inverse Problems in Aerodynamics, Heat Transfer, Elasticity and Materials Design

A number of existing and emerging concepts for formulating solution algorithms applicable to multidisciplinary inverse problems involving aerodynamics, heat conduction, elasticity, and material properties of arbitrary three-dimensional objects are briefly surveyed. Certain unique features of these algorithms and their advantages are sketched for use with boundary element and finite element methods.

Inverse Design of Coolant Flow Passage Shapes with Partially Fixed Internal Geometries

A method is described for the inverse design of complex coolant flow passage shapes in internally cooled turbine blades. This method is a refinement and extension of a method developed by the authors for designing a single coolant hole in turbine blades. The new method allows the turbine designer to specify the number of holes the turbine blade is to have. In addition, the turbine designer may specify that certain portions of the interior coolant flow passage geometry are to remain fixed (eg. struts, surface coolant ejection channels, etc.). Like the original design method, the designer must specify the outer blade surface temperature and heat flux distribution and the desired interior coolant flow passage surface temperature distributions. This solution procedure involves satisfying the dual Dirichlet and Neumann specified boundary conditions of temperature and heat flux on the outer boundary of the airfoil while iteratively modifying the shapes of the coolant flow passages using a least squares optimization procedure that minimizes the error in satisfying the specified Dirichlet temperature boundary condition on the surface of each of the evolving interior holes. Portions of the inner geometry that are specified to be fixed are not modified. A first order panel method is used to solve Laplace’s equation for the steady heat conduction within the solid portions of the hollow blade, making the inverse design procedure very efficient and applicable to realistic geometries. Results are presented for a realistic turbine blade design problem.Copyright