Abstract
Accuracy and performance are key issues for CFD simulation. How to meet the specific accuracy requirements, as well as the optimal simulation performance, is always the research hotspot. This paper presents a general theory of Mesh-Order Independence that is used to guide the configuration of two of the most critical control parameters in a concrete CFD simulation process: grid spacing and discretization order. A concept of optimal mesh-order independent pair which can meet both accuracy and performance requirements at the same time is proposed and analyzed. To find the optimal Mesh-order independent pair, the Mesh-Order Independence is applied to high order FEM simulation, and the specific process and key technologies are detailed. Test and results of two benchmark cases, the Laplace equation and the Helmholtz equation, show that the Mesh-order theory proposed in this paper provides an important guidance for the grid spacing selection and discretization order configuration in practical simulation, especially in the case of high precision requirements. Specifically, only 6 pre-runs with low discretization orders and coarse meshes are needed for the both cases to have a prediction accuracy of more than 70%.
Highlights
As an emerging interdisciplinary, computational fluids dynamics (CFD) uses numerical methods and computer simulations to solve real physical, biological, and chemical problems [1], [2]
It is designed to deal with the trade-offs between the simulation performance and accuracy, by adjusting two critical impact factors, the grid spacing h and discretization order p
In order to demonstrate the specific procedure of choosing h and p based on this theory, a detailed flow chart is given with high-order finite element simulation as an example, and the key technologies involved are described in detail
Summary
Computational fluids dynamics (CFD) uses numerical methods and computer simulations to solve real physical, biological, and chemical problems [1], [2]. B. MESH–ORDER INDEPENDENT PAIR It is known that as the grid spacing decreases, or the discretization order increases, the numerical solution will gradually approach the exact solution, so the error will gradually decrease and the accuracy will be improved. Let e be the error of a CFD simulation with a configuration of pair(h, p), that is, the grid spacing is h and the discretization order is p, for a prescribed threshold ε, if e≤ε is satisfied, the pair(h, p) is called Mesh–order independent pair, denoted as (h, p). Let e be the relative error of a CFD simulation with a configuration of pair(h, p), that is, the grid spacing is h and the discretization order is p, for a prescribed threshold ε, if e ≤ε is satisfied, the Mesh–order pair pair(h, p) relative Mesh–Order Independence, denoted as (h, p). Transitivity If the threshold ε1 < ε2, all the Mesh– order independent pair under the threshold ε1 are the Mesh–order independent pair under the threshold ε2
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