Imposing essential boundary conditions in mesh-free methods

TLDR: This paper presents a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods and special attention is paid to the mesh- free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of theGalerkin weak form.

About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2004-03-26 and is currently open access. It has received 426 citations till now. The article focuses on the topics: Discontinuous Galerkin method & Boundary knot method.

Figures

Fig. 5. Continuous blending of EFG with finite elements: particles are marked with × and nodes are marked with o

Fig. 6. Solution (left) and absolute value of the error (right) for the Lagrange multiplier method

Fig. 14. Solution using boundary linear finite elements for the interpolation of the Lagrange multiplier

Fig. 15. Nitsche’s solution with a 7 × 7 distribution of particles for β = 10 (left), β = 100 (center) and β = 104 (right)

Fig. 4. Detail of a discretization for the continuous blending method with active finite element nodes along the boundary (•) and particles in the interior (×)

Fig. 3. Discretization with active finite element nodes at the boundary (o) and particles (x), and mesh-free shape function associated to the particle located at the gray circle (A) and (B) respectively.

Frequently asked questions

Q1. What is the advantage of Nitsche’s method?

The great advantage of Nitsche’s method is that parametric tuning can be done with only one scalar parameter β, in front of the difficult choice of the interpolation space for the Lagrange multiplier. 

Q2. What is the optimal convergence rate for the penalty parameter?

If the penaltyparameter is constant, or proportional to h−1, the boundary error dominates and the optimal convergence rate is lost as h goes to zero. 

Q3. What is the role of a penalty parameter in the Lagrange multiplier?

For increasing values, β plays the role of a penalty parameter, giving more weight to the verification of the boundary condition and, therefore, affecting to the solution in the rest of the domain. 

Q4. What is the common use of transformation methods?

Transformation methods are usually employed in transient problems, or evolution problems, and transformation matrices are computedonly once. 

Q5. What is the effect of the dilation parameter on the shape functions?

By introducing an extension of the dilation parameter at each particle, the mesh-free shape functions can be forced to verify the Kronecker delta property at the boundary [10]. 

Q6. What is the advantage of a finite element surface mesh?

For instance, in contact problems the interest of a finite element surface mesh is obvious and the advantage of enriching the area close to the boundary is crucial both for precision and large deformations. 

Q7. What is the common choice for the Lagrange multiplier?

The discretization of the multiplier λ must be accurate enough in order to obtain an acceptable solution, but the resulting system of equations turns out to be singular if the number of Lagrange multipliers λi is too large. 

Q8. What is the simplest way to determine the reproducibility of the mesh-free method?

Given a set of particles xi in the domain Ω ⊂ Rn, mesh-less methods are based in a functional interpolation of the formu(x) ' uρ(x) = ∑ i Nρi (x)ui. 

Q9. What is the case of discretization (c)?

This is the case of discretization (c) which corresponds to double the density of collocation points along the essential boundary. 

Q10. What is the penalty parameter for the Lagrange multiplier?

That is,u = arg min v∈H1(Ω)Π(v) + 12 β∫Γd(v − ud)2 dΓ. (18)The penalty parameter β is a positive scalar constant that must be large enough in order to impose the essential boundary condition with the desired accuracy. 

Q11. What is the main difficulty in the interpolation of the multiplier?

A simple 2D linear elasticity problem shows the major difficulties in the practical choice of the interpolation of the multiplier in particular situations. 

Q12. What is the solution of problem (6)?

The solution of problem (6) can also be obtained as the solution of a minimization problem with constraints: “u minimizes the energy functionalΠ(v) = 12∫Ω ∇v · ∇v dΩ−∫Ω vf dΩ−∫Γn vgn dΓ, (12)and verifies the essential boundary conditions.” 

Q13. What is the weak form of the linear elasticity problem?

Nitsche’s weak form of this linear elasticity problem is∫Ω ε(v) : σ(u) dΩ−∫Γd(v ·n)(n ·σ(u) ·n) dΓ− ∫Γd(u ·n)(n ·σ(v) ·n) dΓ+ β ∫Γd(v · n)(u · n) dΓ = − ∫Γdgd(n · σ(v) · n) dΓ + β ∫Γdgd(v · n) dΓfor all v ∈ [H1(Ω)]nsd, where ε(v) is the strain tensor associated to the dis-placement v, and β is a large enough constant which ensures the coercivity of the bilinear form.