A class of nonsymmetric preconditioners for saddle point problems

For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric

On the resolution of the viscous incompressible flow for various SUPG finite element formulations

European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona 11-14 september 2000[Abstract] A Finite Element based program has been released to solve the steady 2D Navier- Stokes equations. The mixed-variable algorithm is used as a first approach to solve the differential problem. In order to reduce the number of equations, both a penalty and segregated formulation are implemented to give solution to the viscous incompressible flow and their results are compared with the former formulation. The program makes use of a SUPG type algorithm as a stabilisation procedure, in order to eliminate the numerical oscillations, which may appear when the boundary conditions force a sudden change in the solution, without necessarily refining the mesh. The three different algorithms are checked making use of the benchmark problems of the flow in a square cavity, the backward step and the flow past a cylinder. Finally the code is used to solve the flow in some practical problems and their results are commented

Finite element implementation of two‐equation and algebraic stress turbulence models for steady incompressible flows

The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two‐equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution

Design for Additive Manufacturing of Conformal Cooling Channels Using Thermal-Fluid Topology Optimization and Application in Injection Molds

Additive manufacturing allows the fabrication parts and tools of high complexity. This capability challenges traditional guidelines in the design of conformal cooling systems in heat exchangers, injection molds, and other parts and tools. Innovative design methods, such as network-based approaches, lattice structures, and structural topology optimization have been used to generate complex and highly efficient cooling systems; however, methods that incorporate coupled thermal and fluid analysis remain scarce. This paper introduces a coupled thermal-fluid topology optimization algorithm for the design of conformal cooling channels. With this method, the channel position problem is replaced to a material distribution problem. The material distribution directly depends on the effect of flow resistance, heat conduction, as well as forced and natural convection. The problem is formulated based on a coupling of Navier-Stokes equations and convection-diffusion equation. The problem is solved by gradient-based optimization after analytical sensitivity derived using the adjoint method. The algorithm leads a two -dimensional conceptual design having optimal heat transfer and balanced flow. The conceptual design is converted to three-dimensional channels and mapped to a morphological surface conformal to the injected part. The method is applied to design an optimal conformal cooling for a real three dimensional injection mold. The feasibility of the final designs is verified through simulations. The final designs can be exported as both three-dimensional graphic and surface mesh CAD format, bringing the manufacture department the convenience to run the tool path for final fitting

Robust Preconditioners for Incompressible MHD Models

In this paper, we develop two classes of robust preconditioners for the structure-preserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners