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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.
The main feature of isogeometric analysis is the usage of a coherent geometry description for design and analysis. In most cases, Non-Uniform Rational B-splines (NURBS) are used within the frame of the finite element method, which combines the most common geometrical modeling concept with the most common analysis method in structural mechanics. NURBS are a versatile tool for geometric modeling, and in order to define complex geometric structures, a multitude of tensor-product NURBS patches is required. Typical software in Computer-Aided design can manage smoothness requirements across common interfaces of neighboring patches. For the finite element method, a different type of coupling across interfaces must be achieved. Due to the nature of tensor product NURBS, the simple concept of mutual refinement and subsequent coupling by shared degrees of freedom is prohibitively costly and, in some cases, even not possible. A multitude of coupling methods has been proposed over the last years. The most common concepts are known as mortar methods. In particular, the dual mortar method has been shown to yield very efficient computations. A recent paper by the authors has provided an isogeometric mortar method with mathematically proven optimal convergence of the stress errors over the entire domain. We use dual basis functions, which have support only on one interface and avoid interrelations between different interfaces. Models with a large number of intersecting interfaces can be handled. However, the basis functions have full support on the interfaces. In our current contribution, we propose the use of approximate dual basis functions with the advantage of having local support on the interfaces. These functions fulfill the duality only in an approximate way, but still guarantee the optimal degree for the convergence of the mortar method. Since the duality is not fulfilled, an additional lumping of the mortar matrix is introduced. The error of this lumping can be analyzed mathematically and is not significant in comparison to the global approximation error of the finite element method. The use of the approximate dual basis functions restores the local support of basis functions along the interface while the convergence properties remain intact. Numerical examples show the convergence behavior for simple and complex models.
Interpolation of Rotations and Coupling of Patches in Isogeometric Reissner– Mindlin Shell Analysis
(2015)
This work is concerned with the development of an efficient and robust isogeometric Reissner–Mindlin shell formulation. The basic assumption of shell theories is a dimensional reduction of the three-dimensional continuum to a two-dimensional surface embedded in the three-dimensional space. Consequently, the geometry is described by a reference surface in combination with a director vector field, which defines the expansion in the thickness direction. The main objective of isogeometric analysis is to use the same model description for design and analysis. Thin-walled structures are usually defined by a reference surface and an associated thickness in industrial design software. Thus, the usage of isogeometric shell elements can avoid costly conversions to volumetric geometry descriptions.The usage of NURBS surfaces (Non-Uniform Rational B-splines) possibly yields high continuity between elements. This requires a rethinking of all concepts used in conventional shell elements, which base on linear Lagrange basis functions. The shell formulation presented in this work is derived from the continuum theory and uses an orthogonal rotation described by Rodrigues' tensor to compute the current director vector. Large deformations and finite rotations can be described accurately. The discretization requires nodal director vectors which interpolate the normal vector as exact as possible. A new method for the definition of nodal basis systems and nodal director vectors is derived. Basing on this, a criterion for the automatic assignment of the correct number of rotational degrees of freedom for each node is proposed. This allows stable computations of geometries with kinks while requiring neither the usage of drilling rotation stabilization nor manual user interaction. The main part of this work is the derivation of various concepts for the interpolation of the current director vector, which is a function of the rotational state. The respective concepts differ in the quantity which is actually interpolated and in the chosen update formulation for the rotations. The influence of each concept on the global deformation convergence behavior is assessed with the help of numerical examples. The results suggest that proper convergence behavior for all orders of NURBS basis functions can only be attained if interpolated director vectors are rotated. Concepts of this type are more accurate and expensive than concepts which rotate nodal director vectors. But the higher computational effort pays off for geometries with arbitrary curvature and for basis functions of higher order. Geometries with kinks require a multiplicative rotational update formulation for concepts that rotate interpolated director vectors.Three different integration rules are considered within the numerical examples. Besides full and reduced Gauss integration also a new non-uniform Gauss integration concept following Adam et al. (2015) is assessed. A special focus is put on the interaction between the chosen rotational concept and the integration scheme. The reduction of the number of integration points from full to reduced integration slightly alleviates locking effects. The further reduction entailed by non-uniform integration significantly reduces locking effects in some examples. But this only yields higher accuracy if a concept which rotates interpolated director vectors is chosen. The reduction of locking deteriorates the accuracy of deformation results in other cases. The efficiency of the presented shell formulation is compared to standard shell formulations in terms of computational costs to attain a pre-defined error level. The most effective combination of integration scheme and rotational concept is shown to be competitive to standard shell formulations.A further main concern of this work is the derivation of a mortar-type method for the coupling of non-conforming NURBS surface patches. Methods to handle non-conforming patches without mutual refinement are essential for an efficient application of NURBS-based isogeometric analysis. The proposed method bases on a substitution relation, which is derived from the weak fulfillment of the equality of mutual displacements along the interface. A static condensation can be performed with the help of this substitution relation in order to attain a coupled global system of equations. The variational formulation is not altered and the global stiffness matrix remains positive definite. Numerical examples show the applicability of the method. A comparison to reference results and to computations with the Lagrange multiplier method is given. The applicability of the coupling method for the presented Reissner-Mindlin shell formulation is shown with the help of two nonlinear examples.
Die isogeometrische Methode definiert sich durch den Einsatz einer einheitlichen Geometriebeschreibung für Entwurf und Berechnung.
Insbesondere bei der Berechnung dünnwandiger Flächentragwerke ist durch die Verwendung der exakten Geometrie ein großer Gewinn an Genauigkeit und Zuverlässigkeit möglich. Um neben hoher Genauigkeit auch hohe Effizienz zu erreichen, wird in diesem Beitrag die Verwendung effizienter numerischer Integrationsmethoden für die Berechnung der Steifigkeitsmatrix untersucht.
Employing isogeometric analysis (IGA), the geometry of structures is discretized by non-uniform rational B-splines (NURBS), which simultaneously provide the basis functions for the corresponding analysis as well. Due to the high continuity within patches, a unique local convective basis system, which corresponds to the local geometry directions of the mesh, can be defined in every point.
Locking Phenomena, which strongly affect purely displacement-based low order elements, can be counteracted by employing particular methods or higher polynomial degrees; both options increase the resulting computational effort significantly. Using order elevation only in specific directions has the potential to optimize the ratio between locking counteraction and computational costs.
In this contribution, the use of directed deformations based on convective basis systems in each control point is proposed for a displacement-based isogeometric formulation with specifically adapted orders. Therefore, distinct meshes for the interpolation of the displacements in each direction are generated based on the initial geometry. Subsequently, the order of every mesh is elevated in only one direction. Consequently, different possibilities for the combination of order elevations have to be examined.
This procedure is conducted for a two-dimensional linear elasticity problem. The benefit of a convective formulation with direction dependent degrees is shown by a comparison to a standard isogeometric formulation. Furthermore, the impact of the directions selected for order elevation on the accuracy of the results is investigated.
We develop a mixed geometrically nonlinear isogeometric Reissner–Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach. The starting point of the formulation is the modified Hellinger–Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the orthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly.
Patch coupling in isogeometric analysis of solids inboundary representation using a mortar approach
(2020)
This contribution is concerned with a coupling approach for nonconforming NURBS patches in the framework of an isogeometric formulation for solids inboundary representation. The boundary representation modeling technique inCAD is the starting point of this approach. We parameterize the solid according to the scaled boundary finite element method and employ NURBS basis functions for the approximation of the solution. Therefore, solid surfaces consist of several sections, which can be regarded as patches and discretized independently. The main objective of this study is to derive an approach for the connection of independent sections in order to allow for local refinement and thus an accurate and efficient discretization of the computational domain. Non-conforming sections are coupled with a mortar approach within a master-slave framework. The coupling of adjacent sections ensures the equality of mutual deformations along the interface in a weak sense and is enforced by constraining the NURBS basis functions on the interface. We apply this approach to non-linear problems in two dimensions and compare the results with conforming discretizations.
An isogeometric Reissner–Mindlin shell derived from the continuum theory is presented. The geometry is described by NURBS surfaces. The kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system. Hence, the definition of nodal basis systems at the control points is necessary for the proposed formulation. The control points are in general not located on the shell reference surface and thus, several choices for the nodal values are possible. The proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof. Thus, the initial director vector coincides with the normal vector even for the coarsest mesh. In addition to that a more accurate interpolation of the current director and its variation is proposed. Instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations. Account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation. The exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes. The convergence behavior is shown to be correct for k-refinement allowing the use of coarse meshes with high orders of NURBS basis functions. This is potentially advantageous for applications with high numerical effort per integration point. The geometrically nonlinear formulation accounts for large rotations. The consistent tangent matrix is derived. Various standard benchmark examples show the superior accuracy of the presented shell formulation. A new benchmark designed to test the convergence behavior for free form surfaces is presented. Despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound.
A NURBS based hybrid collocation-Galerkin method for the analysis of boundary represented solids
(2015)
The paper is concerned with a new numerical method, NURBS based hybrid collocation–Galerkin method (NURBS-HCGM), to solve the in-plane motion problem of elastic solids. It combines the merits of the so-called scaled boundary finite-element method (SB-FEM) and the isogeometric collocation method. For the analysis, the boundary scaling technique of SB-FEM is adopted. It leads to a formulation, where only the boundary of a structure is discretized. Here, the NURBS basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary. This is in accordance with the boundary representation modeling technique, which is commonly employed in computer aided design software. The inner domain is described by a radial scaling parameter. Applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation (ODE) of Euler type, where the unknown displacements are a function of the radial scaling parameter. In the present work a NURBS based collocation scheme is introduced to solve this equation. NURBS basis functions are suggested for the approximation of the displacements in scaling direction. The higher continuity provided by NURBS allows to use collocation to solve the ODE directly instead of using the weak form in scaling direction. The proposed approach is validated by comparison with the eigenvalue solution of the ODE. It is remarked that the eigenvalue solution is restricted to linear problems, whereas the proposed method could be extended to nonlinear problems. In general, the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries. The accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions. The computational cost is investigated with the help of numerical examples and is compared to isogeometric Galerkin approach.
Transverse shear locking is an issue that occurs in Reissner–Mindlin plate and shell elements. It leads to an artificial stiffening of the system and to oscillations in the stress resultants for thin structures. The thinner the structure is, the more pronounced are the effects. Since transverse shear locking is caused by a mismatch in the approximation spaces of the displacements and the rotations, a field-consistent approach is proposed for an isogeometric degenerated Reissner–Mindlin shell formulation. The efficiency and accuracy of the method is investigated for benchmark plate and shell problems. A comparison to element formulations with locking alleviation methods from the literature is provided.