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Institute
In this paper we develop the isogeometric Bézier dual mortar method. It is based on Bézier extraction and projection and is applicable to any spline space which can be represented in Bézier form (i.e., NURBS, T-splines, LR-splines, etc.). The approach weakly enforces the continuity of the solution at patch interfaces and the error can be adaptively controlled by leveraging the refineability of the underlying slave dual spline basis without introducing any additional degrees of freedom. As a consequence, optimal higher-order convergence rates can be achieved without the need for an expensive shared master/slave segmentation step. We also develop weakly continuous geometry as a particular application of isogeometric Bézier dual mortaring. Weakly continuous geometry is a geometry description where the weak continuity constraints are built into properly modified Bézier extraction operators. As a result, multi-patch models can be processed in a solver directly without having to employ a mortaring solution strategy. We demonstrate the utility of the approach on several challenging benchmark problems.
The combination of materials with either pronounced ferroelectric or ferromagnetic effect characterizes multiferroic heterostructures, whereby the different materials can be arranged in layers, columns or inclusions. The magnetization can be controlled by the application of electrical fields through a purely mechanical coupling at the interfaces between the different materials. Thus, a magneto-electric coupling effect is obtained. Within a continuum mechanics formulation, a phase field is used to describe the polarization and the magnetization in the ferroelectric and ferromagnetic layers, respectively. The coupling between polarization/magnetization and strains within the layers, in combination with the mechanical coupling at the sharp layer interfaces, yields the magneto-electric coupling within the heterostructure. The continuum formulations for both layers are discretized in order to make the differential equations amenable to a numerical solution with the finite element method. A state-of-the-art approach is used for the ferroelectric layer. The material behavior of the ferromagnetic layer is described by a continuum formulation from the literature, which is discretized using a newly proposed approach for the consistent interpolation of the magnetization vector. Four numerical examples are presented which show the applicability of the newly proposed approach for the ferromagnetic layer as well as the possibility to simulate magneto-electric coupling in multiferroic heterostructures.
We propose an electro-mechanically coupled phase-field model for ferroelectric materials that show cubic–tetragonal phase transition. The cubic phase is idealized by an isotropic formulation, and the tetragonal phase is idealized by a transversely isotropic formulation. We consider a classical phase-field model with Ginzburg–Landau-type evolution of the order parameter. The order parameter drives the transition of all involved moduli tensors such as elastic, dielectric and piezoelectric moduli, which in turn maintain their typical features and stability as a result of a selected phase-transition function. The model is described in coordinate-invariant form and implemented into a finite element framework with implicit time integration of the evolution equation. Representative numerical examples in two and three dimensions demonstrate the main features of the constitutive model and the numerical stability of the formulation.
This contribution defines and compares different methods for the computation of dual basis functions for B-splines and Non-Uniform Rational B-splines (NURBS). They are intended to be used as test functions for the isogeometric mortar method, but other fields of application are possible, too. Three different concepts are presented and compared. The first concept is the explicit formula for the computation of dual basis functions for NURBS proposed in the work of Carl de Boor. These dual basis functions entail minimal support, i.e., the support of the dual basis functions is equal to the support of the corresponding B-spline basis functions. In the second concept dual basis functions are derived from the inversion of the Gram matrix. These dual basis functions have global support along the interface. The third concept is the use of approximate dual basis functions, which were initially proposed for the use in harmonic analysis. The support of these functions is local but larger than the support of the associated B-spline basis functions. We propose an extension of the approximate dual basis functions for NURBS basis functions. After providing the general formulas, we elaborate explicit expressions for several degrees of spline basis functions. All three approaches are applied in the frame of the mortar method for the coupling of non-conforming NURBS patches. A method which allows complex discretizations with multiple intersecting interfaces is presented. Numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis. This fact is in accordance with mathematical findings in literature, which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions. The dual basis functions computed from the inverse of the Gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes. Only the approximate dual basis functions yield accurate and efficient computations, where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes. All basic cases of T-intersections and star-intersections are studied. Furthermore, an example which combines all basic cases in a complex discretization is given. The applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example.
We discuss the application of non-uniform rational B-splines (NURBS) in the scaled boundary finite element method (SBFEM) for the solution of wave propagation problems at rather high frequencies. We focus on the propagation of guided waves along prismatic structures of constant cross-section. Comparisons are made between NURBS-based discretizations and high-order spectral elements in terms of the achievable convergence rates. We find that for the same order of shape functions, NURBS can lead to significantly smaller errors compared with Lagrange polynomials. The difference becomes particularly important at very high frequencies, where spectral elements are prone to instabilities. Furthermore, we analyze the behavior of NURBS for the discretization of curved boundaries, where the benefit of exact geometry representation becomes crucial even in the low-frequency range.
Isogeometric analysis fosters the integration of design and analysis by using the geometry description of the CAD system also for the numerical analysis. Hereby, the use of NURBS surfaces is common but entails the need for a coupling of non-conforming patches. The use of mortar methods allows a coupling which requires neither additional variables nor empirical parameters. In this contribution dual basis functions are used in order to obtain an accurate and efficient mortar method.
Shell elements for slender structures based on a Reissner-Mindlin approach struggle in pure bending problems. The stiffness of such structures is overestimated due to the transversal shear locking effect.
Here, an isogeometric Reissner-Mindlin shell element is presented, which uses adjusted control meshes for the displacements and rotations in order to create a conforming interpolation of the pure bending compatibility requirement. The method is tested for standard numerical examples.
In this contribution, an isogeometric Reissner-Mindlin shell element is presented, which uses adjusted approximation spaces for the displacements and the rotations in order to avoid transversal shear locking effects. These locking effects arise especially in pure bending problems, with decreasing thickness of the shell. Their origin lies in the not conforming interpolations of the displacements and rotations in the formulation of the compatibility requirements.
One possibility to overcome this difficulty would be to increase the polynomial degree of the used NURBS shape functions, as it is proposed in [1]. However, the locking effects are not completely eliminated and the computational time for the formation of the stiffness matrix increases significantly with rising polynomial degrees. For lower polynomial degrees, there exist only a few effective concepts for the prevention of locking. Beir˜ao da Veiga and his group proposed one of these methods for the elimination of transversal shear locking effects for plate problems [2]. They suggested the implementation of different control meshes, with adjusted polynomial degrees for the interpolation of the displacements and the rotations. In this way transversal shear locking effects due to the coupling of shear strains and curvature are avoided.
The isogeometric concept still holds because the reference geometry for the displacements and the rotations is the same and only the refinement is different.
This method is now extended to an isogeometric Reissner-Mindlin shell formulation. The used shell element is derived from continuum theory. Nodal basis systems are computed with a globalL2 fit and thus the interpolated director vectors of the reference configuration coincide best possible with the normal vectors. The interpolation of the current director vector is performed using a full SO(3) update, so the nodal rotations are interpolated. This deviates from standard rotation interpolation procedures, where the current director vector is interpolated. The more accurate interpolation used in this contribution leads to more accurate results, see [3]. The accuracy and efficiency of the shell element are examined for some standard benchmark examples.
Heterostructures of ferroelectric and ferromagnetic layers are commonly used to obtain electromagnetic effects. The elastic coupling between the layers is widely acknowledged as the main mechanism responsible for the electro-magneto interaction. Within this contribution we study the coupling of ferroelectric and ferromagnetic layers with well-defined interfaces. The intention is to simulate the switching of the magnetization with the help of electric fields, which has been studied experimentally in [1]. Each layer is simulated by using mechanically coupled phase field modeling, whereby the approaches presented in [2] and [3] will be used. The strains in each layer depend on the direction of the polarization/magnetization. A mismatch of these strains will be compensated by local deformations at the interface as the coupling results from the coherent deformation at the interface. This leads to the possibility to alter the magnetization direction by changing the electric polarization and vice verse. Numerical simulations will illustrate the evolution of the ferroic microstructures with a focus on the strain coupling and the resulting interactions between layers.
We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patchesk with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces Xh,k on each patch k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an
extension of Brivadis et al. (Comput Methods Appl Mech Eng 284:292–319, 2015) and highlights several aspects which did not receive full attention before. In particular, by choosing appropriate spaces of polynomial splines as Lagrange multipliers, we obtain
a uniform infsup-inequality. Moreover, we provide a new additional condition on the discrete spaces Xh,k which is required for obtaining optimal convergence rates of the mortar method. Our numerical examples demonstrate that the optimal rate is lost if this condition is neglected.