A unified approach for embedded boundary conditions for fourth‐order elliptic problems

Abstract

An efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier-Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche's approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non-conforming meshes with non-interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B-splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter. 656 I. HARARI AND E. GROSU of freedom. However, the method suffers from the lack of a rigorous criterion for defining its stabilization parameter. A discontinuous Galerkin method [9] alleviates this difficulty. Another approach [10], with deflection and rotation degrees of freedom, is based on a stabilized method for a mixed representation.Smooth functions such as B-splines attain higher global continuity with relative ease, making them suitable for higher-order problems. B-splines of degree k are typically C k 1 continuous. Such bases are becoming increasingly popular for use in finite element computations, in part due to their ability to provide exact geometric representations (as with the isogeometric concept [11]). This has led to renewed interest in direct discretization of C 1 theories such as thin plate (and shell) bending [12][13][14].Conventional finite elements based on virtual work treat kinematic boundary conditions as essential conditions, requiring the basis to be interpolatory and the mesh to conform to the boundary of the domain. Weakly enforcing the kinematic boundary conditions eliminates the geometric limitation of classical C 1 elements such as Bogner-Fox-Schmit. This notion was explored in the context of hybrid methods [15,16]. (Similar ideas are used to enforce inter-element continuity for timeharmonic waves [17].) However, hybrid methods require additional degrees of freedom for the auxiliary fields and, as in the case of mixed methods, are subject to stability issues.We present a variational formulation for fourth-order elliptic problems with embedded Dirichlet boundary conditions based on the Nitsche approach. This is one of several techniques under the broad category of immersed boundary and interface...

A parameter-free variational coupling approach for trimmed isogeometric thin shells

“…For optimal performance of the method, α needs to be chosen as small as possible. Element-wise configuration dependent stabilization parameters can be estimated based on a local eigenvalue problem [40,7,43,44]. The particular method (19) makes use of one-sided inequalities to establish estimates of local stabilization parameters.…”

“…On the embedded domain side, the importance of geometrically faithful quadrature of trimmed elements and corresponding techniques have been discussed in a series of recent papers [28,27,[29][30][31][32][33][34][35]. For the weak enforcement of boundary and interface conditions at trimming curves and surfaces, variational methods such as Lagrange multiplier [36][37][38] or Nitsche techniques [39][40][41][42][43][44] have been successfully developed.…”

“…If they are too small, stability is lost. Moreover, accurate estimation techniques are often delicate from an algorithmic viewpoint [43,44,46]. Therefore, there has been an increasing interest in methods that can enforce boundary and interface conditions without mesh dependent stabilization parameters [38,[47][48][49].…”

Essential boundary and interface conditions in the meshless analysis of shells.

“…The shell speci c internal power per unit volume at the reference con guration can be expressed by 26) where P = τ α ⊗ e r α is the rst Piolla-Kirchho stress tensor, τ r α = Q oT τ α are its backrotated column vectors andḞ is the velocity gradient. Integration across the thickness yields the internal power per unit area of the midsurface in the initial con guration 27) where so that the internal the internal power of the shell is…”

“…It has been applied with eXtended Finite Elements (XFEM) for EBCs and ICs in single-eld problems [15,25,28], to enforce weak discontinuities between elastic materials [51,52,2,1]; and applied to non conforming embedded FE meshes for elastostatics [53,54]. B-splines and NURBs-base FEs share similar features with meshless approximations and can also take advantage of Nitsche's Method, as in [16] for 2nd and 4th order problems, [46,33,49,50] for elasticity, [27,26] for thin plates and [22,23] for thin shells. The theory has also been extended for contact with FEs [59] and XFEM [4,3].…”