FG Statik und Dynamik
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Der Beitrag vergleicht die Genauigkeit von Reissner-Mindlin-Schalenelementen zwischen der isogeometrischen Methode und der spektralen Elemente-Methode. Während die erste durch die hohe Kontinuität zwischen den Elementen eine sehr hohe Genauigkeit in Bezug auf Anzahl der Freiheitsgrade aufweist, besticht die letztere durch eine einfachere Formulierung und eine bessere Konditionierung auch für sehr hohe Ansatzordnungen.
A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition
(2024)
A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.
An efficient mass lumping scheme for isogeometric analysis based on approximate dual basis functions
(2024)
In this contribution, we propose a new mass lumping scheme for explicit dynamics in isogeometric analysis (IGA). To this end, an element formulation based on the idea of dual functionals is developed. Non-Uniform Rational B-splines (NURBS) are applied as shape functions and their corresponding dual basis functions are applied as test functions in the variational form, where two kinds of dual basis functions are compared. The first type are approximate dual basis functions (AD) with varying degree of reproduction, resulting in banded and diagonally-dominant mass matrices. Dual basis functions derived from the inversion of the Gram matrix (IG) are the second type and already yield diagonal mass matrices. We will show that it is possible to apply the dual scheme as a transformation of the resulting system of equations based on NURBS for both – shape and test functions. Hence, it can be easily implemented into existing IGA routines and it is also promising to retain the accuracy known from similar formulations without mass lumping. Applying additional row-sum lumping to the mass matrices is either not necessary for IG or the caused loss of accuracy is lowered to a reasonable magnitude in the case of AD. Numerical examples show a significantly better approximation of the dynamic behavior for the dual lumping scheme compared to standard NURBS approaches making use of conventional row-sum lumping. In a nutshell, applying IG yields accurate numerical results but fully populated stiffness matrices occur, which are entirely unsuitable for explicit dynamic simulations, while combining AD and row-sum lumping leads to efficient dynamical computations, with respect to effort and accuracy.
In order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non-matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner-Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non-uniform rational B-splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero-energy modes.
Für eine Lebensdauervorhersage veränderlich belasteter Tragwerke, etwa des Anlagen- und Maschinenbaus sowie des Bauingenieurwesens, werden die zyklisch akkumulierten Verzerrungen und ggf. auch die elastisch-plastischen Dehnschwingbreiten benötigt. Die Vereinfachte Fließzonentheorie (VFZT) ist eine direkte Methode, die Abschätzungen dieser und aller anderen mechanischen Größen im elastischen und im plastischen Einspielzustand liefert. Das vorliegende Buch stellt die VFZT ausführlich dar und legt Wert darauf, dass sich nicht nur Wissenschaftler, sondern auch in der Praxis tätige Ingenieure sowie Studierende höherer Semester ein Bild von den Möglichkeiten und Grenzen machen können. Zahlreiche Abbildungen und Anwendungsbeispiele unterstützen das Verständnis.
On the use of mixed basis function degrees within a convective isogeometric element formulation
(2023)
Solving linear elasticity problems using standard finite element methods, different locking phenomena can occur. In order to counteract these effects, mixed methods or formulations using higher approximation orders can be employed, for instance. As a consequence, this leads to an increased computational effort. Hence, a selective elevation of orders in decisive directions within a purely displacement-based element formulation is proposed in this contribution. Within isogeometric analysis (IGA), the geometry is discretized using non-uniform rational B-splines (NURBS), which simultaneously represent the basis functions for the analysis. Due to the fact that the geometry can be preserved exactly during analysis, this can increase the accuracy of results. In this contribution, convective basis systems that are aligned with the local geometry are employed combined with selective order elevation. The required convective basis systems are interpolated from those determined in each control point.
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.
The focus of this study is on providing an efficient and highly accurate mass lumping scheme for explicit dynamic calculations in isogeometric analysis (IGA). Computer-Aided design models make use of higher-order polynomials, usually Non-Uniform Rational Splines (NURBS), to build up the geometry model. In IGA they are chosen as shape functions to take advantage of their high continuity and to keep the description of geometry exact.
Thus, compared to standard Finite Element Method (FEM), a smaller number of elements is required to gain results of the same quality level. In general, explicit time integration methods require a huge number of time steps to obtain numerical stable results of reasonable quality. Hence, diagonal mass matrices are preferred to reduce the computational costs within each time step. Row-sum lumping and diagonal scaling method are two well-known techniques, which have been developed as mass lumping schemes for standard FEM.
Unfortunately, these standard FEM mass lumping schemes deteriorate convergence rates in IGA dynamics significantly. With rising order of the basis functions, accuracy of the results decreases. The reason for this is that shape functions of higher order lead to more dense mass matrices and thus the error caused by lumping the mass matrices increases. Therefore, new mass lumping schemes have to be developed, which are more suitable for the use of higher-order shape functions.
Several types of dual basis functions (duals), which have already been used in the isogeometric mortar method, are considered for a new mass lumping scheme. Keeping the initial NURBS as shape functions, the duals are applied as test functions for the variational form. Depending on the chosen type of duals, diagonal or banded mass matrices are obtained without any additional lumping. Compared to lumping the original mass matrices based on NURBS as test functions, applying additional row-sum lumping to the banded matrices causes significantly smaller errors.
The presented new mass lumping scheme is a very promising approach, in which duals are used as test functions in IGA dynamics with explicit time integration, e.g. the Central Differences Method. The implementation into existing codes can be easily done. The assembled system matrices, if based on NURBS as shape and test functions, only have to be multiplied with a transformation matrix. Numerical examples show that if using this dual lumping scheme within IGA methods, explicit dynamic analysis can be performed efficiently with higher-order shape functions.
Many pressure vessel and piping components have to withstand high internal pressures and are therefore thick-walled so that geometric effects such as stress stiffening need not be accounted for. However, thin-walled, or moderately thick structures may be sensitive to these effects. Design Codes such as the ASME Boiler and Pressure Vessel Code usually provide little guidance on when they are to be accounted for. In general, the effects of stress stiffening are difficult to estimate even for experienced engineers and can only be estimated by detailed finite element analyses. In the opinion of the authors, this effect deserves more attention. This is particularly true for simplified elastic-plastic methods for fatigue and ratcheting assessment of structures subjected to cyclic loading.
Cyclic loading may cause elastic-plastic strains to accumulate if a ratcheting mechanism is present. After a number of cycles, strain accumulation may cease so that a state of either elastic or plastic shakedown is reached. Determination of accumulated strains, strain ranges and other quantities in the state of shakedown (post-shakedown quantities) by means of incremental analyses is costly, in particular if geometric effects such as stress stiffening play a role. Direct methods aim at providing estimates of the post-shakedown quantities, bypassing cycle-by-cycle analyses. These methods claim to deliver the post-shakedown quantities with high accuracy and low computational effort.
The Simplified Theory of Plastic Zones can account for the combination of plasticity and stress stiffening. The theory is described and illustrated by examples. The Simplified Theory of Plastic Zones (STPZ) has proven itself for estimating the post-shakedown quantities in the state of elastic and plastic shakedown within the framework of the 1st order theory, i.e. if the equilibrium conditions are satisfied for the undeformed structure.
In this paper, the results of elbows subjected to various loading parameters are compared, considering and neglecting stress stiffening. Thus, the results show the influence 2nd order effects can generate. It is further shown that the STPZ can capture 2nd order effects introduced by the equilibrium of the deformed structure. Some examples are used to demonstrate its applicability and the quality of the results, e.g. for a pipe bend subjected to cyclic in-plane bending.
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.
The behavior of elastic-plastic structures under cyclic loading can be determined by incremental elastic-plastic analyses where a given load histogram is analysed cycle-by-cycle until shakedown is achieved. Many cycles may be required for this, if a ratchet mechanism is causing plastic strains to accumulate in each cycle until either elastic or plastic shakedown is achieved. The complexity of the elastic-plastic response of a structure is further increased if geometric effects are present. It may be very costly to get the accumulated strains and strain ranges in the state of shakedown by incremental analyses so that simplified or direct methods have been developed as an alternative. The Simplified Theory of Plastic Zones (STPZ) has proven itself for estimating the required quantities in the state of elastic and plastic shakedown within the framework of the 1st order theory, i.e., if the equilibrium conditions are satisfied for the undeformed structure. It is shown in this paper, how the STPZ can be expanded to capture 2nd order effects introduced by the equilibrium of the deformed structure. Some examples are used to demonstrate its applicability and the quality of the results, but also its limitations with respect to determining the accumulated strains and elastic-plastic strain ranges.
The aim of this study is to provide a new mass lumping scheme for explicit dynamic calculations in isogeometric analysis (IGA).
Shape functions for IGA methods are taken from the Computer-Aided design (CAD) model. The use of higher-order polynomials, usually Non-Uniform Rational B-Splines (NURBS), ensures an exact geometry description [1]. Thus, in comparison to standard finite element method (FEM) the number of elements and therefore the computational costs can be lowered. In addition to that the convergence rate raises equally with the polynomial order. Increasing the degree of shape functions,
less elements are necessary for results of the same quality.
In explicit dynamics, mass lumping schemes are commonly applied to reduce computational costs by using diagonal mass matrices. Well-known techniques like row-sum lumping or diagonal scaling method were developed for dynamic analysis with standard FEM. Unfortunately, they are neither suitable for higher-order shape functions in IGA, nor in the spectral element method [2]. It is not possible to take advantage of increasing the polynomial order to lower the number of elements
in explicit dynamic IGA, because the error caused by lumping the mass matrix also increases. Thus, other mass lumping schemes, suitable for higher-order shape functions, have to be developed. As they are already used for isogeometric mortar method [3], several types of dual basis functions (duals) are studied for a new mass lumping scheme. Using duals as test functions in IGA leads to diagonal consistent mass matrices [4] or banded matrices, depending on the chosen type of duals.
In case of the banded matrices, additional row-sum lumping causes smaller errors than lumping the original mass matrix with standard NURBS as test functions. For IGA dynamics with explicit time integration, e.g. the Central Difference Method, using duals as test functions is a very promising approach. They can be easily implemented in existing methods through multiplying the already assembled system matrices with a transformation matrix based on the underlying NURBS curve. Applying the dual lumping scheme, an explicit dynamic analysis
could also be performed efficiently using IGA methods with higher-order shape functions.
In isogeometric analysis (IGA), which was founded by Hughes et al. [1], the geometry representation is used for the analysis as well. Hence, due to the exact description of the geometry, analysis results can be improved [1, 2]. Therefore, different kinds of splines, like non-uniform rational B-splines (NURBS) [3], are used as shape functions for the discretizations. In linear elasticity problems, for standard formulations, shear locking phenomena can occur due to the different orders of the derivatives in the unbalanced strain-displacement relation. This effect can be reduced using shape functions of higher order, causing an increased computational effort. For low-order formulations, the degrees of shape functions can be adapted accordingly in order to counteract this effect. For an isogeometric displacement-stress mixed Reissner-Mindlin shell formulation such adapted approximation spaces were investigated in [4], for instance. In [5], additionally to a proper choice of shape function spaces, convective coordinates are employed in the derivation of isogeometric shell formulations.
In this contribution, a convective displacement-based isogeometric formulation is introduced, wherein the displacements in the different surface directions are approximated independently using appropriate approximation orders. Therefore, two different meshes are generated from the NURBSdescribed geometry representation, employing order elevation solely for one of the two surface directions. Thus, the order is elevated in opposite directions for these two meshes. The two different possibilities of order elevation are investigated. Furthermore, the use of different convective basis systems is studied. This comprises convective basis systems in each control point, computed according to [6], as well as convective basis systems determined from the local geometry direction in each integration point. The achieved results are compared to whose of a two-dimensional displacementstress mixed formulation presented in [7].
Isogeometric analysis and mixed finite element methods offer promising opportunities to enhance analysis results for complex problems like incompressible elasticity and are able to cope with different locking phenomena. In this contribution, a mixed two-field isogeometric formulation with independent approximations for displacements and stresses is derived, and its ability to counteract different types of locking is investigated using two examples. Furthermore, the influence of the continuity of the stress shape functions on the accurancy of results and convergence behaviour is shown.
An isogeometric element formulation for linear two-dimensional elasticity based on the Airy equation
(2022)
The aim of this work is to derive a formulation for linear two-dimensional elasticity using just one degree of freedom. This degree of freedom is used to directly discretize the Airy bipotential equation, which requires higher order basis functions. Isogeometric structural analysis is based on shape functions of the geometry description in Computer-Aided design software.
These shape functions can easily fulfill the continuity requirement of the bipotential equation.
Thus, an Airy element formulation can be obtained through isogeometric methods. In this contribution Non-Uniform Rational B-splines are used to discretize the domain and to solve the occurring differential equations. Numerical examples demonstrate the accuracy of the evolved formulation for a quadratic plate under different load situations.
Cyclic loading may cause elastic-plastic strains to accumulate if a ratcheting mechanism is present. After a number of cycles, strain accumulation may cease so that a state of either elastic or plastic shakedown is reached. Determination of strains and other quantities in the state of shakedown (post-shakedown quantities) by means of incremental analyses is costly. Direct methods such as the Simplified Theory of Plastic Zones (STPZ) aim at providing estimates of the post-shakedown quantities, bypassing cycle-by-cycle analyses. If geometric effects such as stress stiffening play a role, determination of accumulated strains even becomes more complicated. The STPZ has been further developed in order to account for the combination of plasticity and stress stiffening with respect to elastic shakedown. The theory is described and illustrated using examples such as a pipe bend subjected to cyclic in-plane bending. The implications of cyclic as opposed to constant stress stiffness are discussed. The effect of stress stiffening on ratcheting interaction diagrams (RID), separating regions of elastic and plastic shakedown in the space of loading parameters, is discussed.
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.