A Polygonal Finite Element Method for Stokes Equations

In this paper, we extend the Bernardi-Raugel element [1] to convex polygonal meshes by using the generalized barycentric coordinates. Comparing to traditional discretizations defined on triangular and rectangular meshes, polygonal meshes can be more flexible when dealing with complicated domains or domains with curved boundaries. Theoretical analysis of the new element follows the standard mixed finite element theory for Stokes equations, i.e., we shall prove the discrete inf-sup condition (LBB condition) by constructing a Fortin operator. Because there is no scaling argument on polygonal meshes and the generalized barycentric coordinates are in general not polynomials, special treatments are required in the analysis. We prove that the extended Bernardi-Raugel element has optimal convergence rates. Supporting numerical results are also presented.

Leak-Before-Break Assessment of a Cracked Reactor Vessel Nozzle Using 3D Crack Meshes

A postulated surface crack near a reactor pressure vessel nozzle is evaluated using finite element analysis (FEA) to compute the fatigue crack growth rate, evaluate crack stability, and examine the possibility of a leak-before-break (LBB) condition. For a pressurized vessel with cyclic loading, determining if the crack may have a LBB condition is desirable to allow for the possibility of leak detection leading to corrective action before catastrophic failure. A fatigue crack growth analysis is used to determine how the surface crack dimensions develop before re-categorizing the surface crack as a through thickness crack and evaluating its stability for LBB. To evaluate if a particular crack is unstable and may cause a structural failure, the Failure Assessment Diagram (FAD) method provides an evaluation using two ratios: brittle fracture and plastic collapse. The FAD method is described in the engineering best practice standard API 579-1/ASME FFS-1. The FAD curve and assessment ratios can be obtained from crack front J-integral values, which are computed using 3D crack meshes and elastic and elastic-plastic FEA. Computing custom crack solutions is beneficial when structural component geometries do not have an available stress intensity or reference stress solution.

On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

The Ladyženskaja–Babuška–Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.