Exact imposition of essential boundary condition and material interface continuity in Galerkin-based meshless methods

2016 ◽  
Vol 110 (7) ◽  
pp. 637-660 ◽  
Author(s):  
Hong Zheng ◽  
Wei Li ◽  
Xiuli Du
Keyword(s):  
Boundary Condition ◽  
Meshless Methods ◽  
Material Interface ◽  
Essential Boundary Condition

The Characteristics-Based Matching (CBM) Method for Compressible Flow With Moving Boundaries and Interfaces

2004 ◽  
Vol 126 (4) ◽  
pp. 586-604 ◽  
Author(s):  
R. R. Nourgaliev ◽  
T. N. Dinh ◽  
T. G. Theofanous
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Level Set ◽  
Compressible Flows ◽  
Radical Change ◽  
Subsequent Paper ◽  
Ghost Fluid Method ◽  
Material Interface ◽  
Interface Position ◽  
Level Set Function

Recently, Eulerian methods for capturing interfaces in multi-fluid problems become increasingly popular. While these methods can effectively handle significant deformations of interface, the treatment of the boundary conditions in certain classes of compressible flows are known to produce nonphysical oscillations due to the radical change in equation of state across the material interface. One promising recent development to overcome these problems is the Ghost Fluid Method (GFM). The present study initiates a new methodology for boundary condition capturing in multifluid compressible flows. The method, named Characteristics-Based Matching (CBM), capitalizes on recent developments of the level set method and related techniques, i.e., PDE-based re-initialization and extrapolation, and the Ghost Fluid Method (GFM). Specifically, the CBM utilizes the level set function to capture interface position and a GFM-like strategy to tag computational nodes. In difference to the GFM method, which employs a boundary condition capturing in primitive variables, the CBM method implements boundary conditions based on a characteristic decomposition in the direction normal to the boundary. In this way overspecification of boundary conditions is avoided and we believe so will be spurious oscillations. In this paper, we treat (moving or stationary) fluid-solid interfaces and present numerical results for a select set of test cases. Extension to fluid-fluid interfaces will be presented in a subsequent paper.

scholarly journals Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

2019 ◽  
Vol 53 (3) ◽  
pp. 869-891
Author(s):  
Takahito Kashiwabara ◽  
Issei Oikawa ◽  
Guanyu Zhou
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Trace Operator ◽  
Essential Boundary Condition ◽  
Slip Boundary ◽  
Polyhedral Domain ◽  
Discrete Counterpart ◽  
Original Domain

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ ℝN (N = 2,3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n∂Ω = g on ∂Ω. Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation of n∂Ω by n∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H1 (Ω)N → H1/2(∂Ω); u ↦ u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(hα + ε) and O(h2α + ε) for the velocity in the H1- and L2-norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.

Development of the Smoothed Natural Neighbor Petrov-Galerkin Method

2012 ◽  
Vol 201-202 ◽  
pp. 198-201
Author(s):  
Kai Wang ◽  
Shen Jie Zhou ◽  
Zhi Feng Nie
Keyword(s):  
Boundary Condition ◽  
Galerkin Method ◽  
Accurate Result ◽  
Computational Cost ◽  
Smoothing Technique ◽  
Essential Boundary Condition ◽  
Natural Neighbor ◽  
Strain Smoothing ◽  
Mlpg Method ◽  
Domain Integration

The strain smoothing technique is employed in the natural neighbor Petrov-Galerkin method (NNPG), and the so-called smoothed natural neighbor Petrov-Galerkin method is proposed and studied. This method inherits the advantages of the generalized MLPG method and possesses the easy imposition of essential boundary condition and the domain integration is completely avoided. In comparison with the traditional NNPG, the smoothed natural neighbor Petrov-Galerkin method can obtained more stable and accurate result without increasing the computational cost.

Sign in / Sign up

Export Citation Format

Share Document