Skew-symmetric Nitsche’s formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact

Abstract

A simple skew-symmetric Nitsche's formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary or interface conditions, the skew-symmetric formulation is parameter-free.For contact conditions, it remains stable and accurate for a wide range of the stabilization parameter.Several numerical tests are performed to illustrate its accuracy, stability and convergence performance. We investigate particularly the effects introduced by Nitsche's coupling, including the convergence performance and condition numbers in statics as well as the extra "outlier" frequencies and corresponding eigenmodes in structural dynamics. We present the Hertz test, the block test, and a 3D self-contact example showing that the skew-symmetric Nitsche's formulation is a suitable approach to simulate contact problems in IGA.have the ability to exactly describe geometries: thus, no geometrical approximation error is introduced.Moreover NURBS are widely adopted in commercial computer-aided design (CAD) packages, and this CAD data can directly be used to construct approximations. In boundary element method (BEM), this translates into the ability to solve directly from the field variables at the control points defining the geometry [78,77,74,59,10,60,58,68]. In FEM, a 3D parameterization of the volume is still necessary [87,88], except when solving shell-like problems [53,13,14,38,49]. The present paper focuses on two following issues. One first issue in IGA is related to boundary conditions, especially essential boundary conditions. Indeed, since NURBS are non-interpolatory, enforcing boundary conditions and constraints cannot be done as simply as in Lagrange FEM: they require tackling difficulties which are similar to those encountered in meshless methods [66] and implicit/immersed boundary methods [37,46]. One second issue in IGA comes from interface conditions and patch coupling: for complex geometries, patch-wise CAD modeling is necessary, and transmission conditions need to be satisfied. The same also arises when gluing heterogeneous materials.Various methods already exist to treat boundary or interface conditions weakly, that have been firstly designed for instance in the FEM context. They are applicable, or have already been applied, for IGA.The most widespread ones are the penalty method, mixed/mortar methods and Nitsche's method. The penalty method [7, 54] is simple but not consistent. Therefore the value of the penalty parameter has to be chosen with great care to achieve the best balance between accuracy and stability. As a matter of fact, if the penalty parameter is chosen too small the boundary or interface conditions are imposed inaccurately, whereas if it is chosen much larger than needed the penalized problem becomes ill-conditioned. Mixed methods for boundary conditions [6] introd...

Design of metamaterial-based heat manipulators by isogeometric shape optimization

“…Here the objective is to force the temperature continuity in a stricter sense and normal flux continuity in a weaker sense. In the literature [30,31], it is also reported that the large stabilization parameter might cause ill-conditioning of the system, but we did not face any conditioning issue for our boundary value problem. For the averaging parameter, a standard value γ = 0.5 is taken, which gives equal weights to fluxes on both sides of the interface.…”

“…In addition to that, it keeps the coercivity of the bilinear form intact and the variational form consistent. Nitsche's method has been successfully applied for patch coupling in [30,31]. When Nitsche's method is applied to couple patches, the linear form (Equation ( 6)) on the right side remains the same, while the bilinear form (Equation ( 5)) is altered as follows, (8) where n is the normal at Γ I for any one patch from the patches connected at Γ I (n = n 1 = − n 2 ), β is the stabilization parameter.…”

Hybrid nearly singular integration for three-dimensional isogeometric boundary element analysis of coatings and other thin structures

“…These difficulties are: (i) the performance of local refinement [11], (ii) an automatic parameterisation to build the approximation, (iii) enforcement of multi-patch constraints, and (iv) the need for an interior parameterisation to be constructed from the CAD data, which only provides boundary information [12]. Several methods were later devised in order to alleviate the difficulties faced by IGA [13][14][15]. And more details about IGA can be found in review papers [16][17][18][19].…”

Isogeometric analysis of thin Reissner–Mindlin shells: locking phenomena and B-bar method

“…The Kirchhoff-Love type elements are rotation-free and are only valid for thin structures. Due to the absence of rotational degrees of freedom (DoFs), special techniques are required to deal with the rotational boundary conditions [9,15,16] and multi-patch connection [17]. Theoretically, the Reissner-Mindlin theory is valid for both thick and thin structures, however it is observed from the literature [11,18] that both the FEM and the IGA approaches suffer from locking for thin structures when the kinematics is represented by Reissner-Mindlin theory, especially for lower order elements and coarse meshes.…”