Position innerhalb des Seitenbaumes

Startseite
Forschung
Intrinsisch lockingfreie Formulierungen

Intrinsisch lockingfreie Formulierungen

Forschungsprojekt

Eine Studie über geometrisch und materiell lockingfreie Finite-Elemente-Formulierungen

Überblick

Anwendbarkeit hierarchischer Formulierungen und Mixed-Displacement-Formulierungen für strukturmechanisch Probleme
Hierarchische Reparametrisierungstechniken für Membranlocking
Eliminierung intrinsisch volumetrischer Lockingeffekte

Projektbeschreibung

Hintergrund

Locking führt bei Problemen der Strukturmechanik zu einer geringeren Konvergenzrate im präasymptotischen Bereich. Locking-Effekte hängen von bestimmten kritischen Parametern ab, z. B. der Schlankheit einer Schale. Dieses numerische Problem führt zu unterschätzten Verschiebungswerten und oszillierenden, parasitären Spannungen. Locking tritt unabhängig vom Diskretisierungsansatz (wie der Finite-Elemente-Methode (FEM), der isogeometrischen Analyse (IGA), netzfreien Methoden oder Kollokationsmethoden) auf, welcher zur Lösung der zu Grunde liegenden physikalischen Gleichungen verwendet wird. Obwohl im Laufe der Jahre viele Lösungen zur Beseitigung oder Abminderung von Locking-Effekten entwickelt wurden (z. B. Enhanced-Assumed-Strain- (EAS) und Assumed-Natural-Strain-Methode (ANS), reduzierte Integration, hybride Methoden, Discrete-Strain-Gap-Methode), sind diese meist spezifisch für ein Diskretisierungsverfahren (z. B. die FEM) und nicht direkt auf andere Diskretisierungsmethoden übertragbar. Der Ursprung von Locking ergibt sich jedoch nicht aus dem Diskretisierungsschema, sondern aus den Differentialgleichungen, die das eigentliche physikalische Problem bestimmen. Dies führt zu der Motivation, das zu steife Verhalten auf einer theoretischen Ebene offenzulegen.

Die Philosophie besteht darin, die Gründe für Locking zu beseitigen, anstatt die Symptome zu bekämpfen.

Hierarchische Formulierungen und eine Mixed-Displacement-Formulierung

Ziel ist es, ein intrinsisch lockingfreies Problem zu erhalten, das mit jedem Standard-Diskretisierungsschema gelöst werden kann und ohne besondere Tricks eine grobmaschige Genauigkeit unabhängig von den Problemparametern erreicht. Aus diesem Grund wurden in den letzten zehn Jahren eine Reihe hierarchischer Formulierungen und eine Mixed-Displacement-Formulierung (MD) entwickelt. Erstere wurde durch Reparametrisierung der maßgeblichen kinematischen Gleichungen gewonnen, während letztere einen speziell entwickelten Differentialoperator enthält, der auf neue unabhängige Variablen angewendet wird, um einen besseren Ansatzraum zu erhalten. Die grundsätzliche Gültigkeit dieser Ansätze wurde bereits mit mehreren numerischen Beispielen bestätigt. Dennoch bringen diese Methoden ihre eigenen Herausforderungen mit sich.

Ergebnisse der Querkräfte (Q_xz) für eine gelenkig gelagerte Platte (10x10) mit gleichmäßig verteilter Last und Schlankheit L/t=10000

Ergebnisse der Querkräfte (Q_xz) in Plattenmitte (y=5.0) für eine gelenkig gelagerte Platte (10x10) mit gleichmäßig verteilter Last und Schlankheit L/t=10000

Herausforderungen und Ziel der Forschung

Für diese neuen Formulierungen ist die Aufbringung von Randbedingungen nicht einfach. Bei hierarchischen Formulierungen führt die Reparametrisierung dazu, dass die gesamte Querdehnung oder neu eingeführte Schubverschiebungen als unabhängige Variablen gewählt werden müssen. Dies erschwert die Anwendung von Rotations-Dirichlet-Randbedingungen. Diese neuen unabhängigen Variablen schaffen auch einen parasitären Nullraum der Steifigkeitsmatrix, der durch die Festlegung bestimmter zusätzlicher Randedingungen entfernt werden muss. Die neu eingeführten unabhängigen Variablen in den Mixed-Displacement-Formulierungen erfordern ähnliche Randbedingungen. Darüber hinaus sind hierarchische Formulierungen besser geeignet, um nur Querschublocking zu vermeiden, während die Mixed-Displacement-Formulierung verwendet werden kann, um sowohl Querschub- als auch Membranlocking zu vermeiden.

Das aktuelle Forschungsprojekt konzentriert sich auf die Untersuchung der zusätzlich erforderlichen zusätzlichen Randbedingungen mit der Motivation einer allgemeineren Anwendbarkeit dieser Formulierungen. Techniken zur Reparametrisierung werden weiter erforscht, um auch Membranlocking-Effekte zu vermindern. Ein weiterer Schwerpunkt des Forschungsprojekts ist die Untersuchung der Möglichkeit, ähnliche Methoden zu entwickeln, um volumetrische Lockingeffekte abzuschwächen.

Projektdaten

Projekttitel:
Intrinsisch lockingfreie Formulierungen für Probleme der Strukturmechanik
Förderung:
Deutsche Forschungsgemeinschaft (DFG), Sachbeihilfe BI 722/12-1, GEPRIS-Projektnummer 452589815

Veröffentlichungen

Oesterle, B., Thierer, R., Krauß, L.-M., & Bischoff, M. (2024). Hierarchische Formulierungen für statische und dynamische Analysen von Flächentragwerken. In B. Oesterle, A. Bögle, W. Weber, & L. Striefler (Hrsg.), Berichte der Fachtagung Baustatik – Baupraxis 15, 04. und 05. März 2024, Hamburg (S. 357--364). https://doi.org/10.15480/882.9247
- Zusammenfassung
- BibTeX
Zusammenfassung
Hierarchische Balken-, Platten- und Schalenformulierungen basieren auf einer geschickten Reparametrisierung der kinematischen Gleichungen, die sich für neuartige, glatte Diskretisierungsverfahren als vorteilhaft erweist. Im vorliegenden Beitrag werden die intrinsischen Eigenschaften des hierarchischen Konzepts anhand statischer und dynamischer Analysen von Flächentragwerken aufgezeigt und diskutiert.
BibTeX
@inproceedings{oesterle2024hierarchische, abstract = {Hierarchische Balken-, Platten- und Schalenformulierungen basieren auf einer geschickten Reparametrisierung der kinematischen Gleichungen, die sich für neuartige, glatte Diskretisierungsverfahren als vorteilhaft erweist. Im vorliegenden Beitrag werden die intrinsischen Eigenschaften des hierarchischen Konzepts anhand statischer und dynamischer Analysen von Flächentragwerken aufgezeigt und diskutiert.}, author = {Oesterle, Bastian and Thierer, Rebecca and Krauß, Lisa-Marie and Bischoff, Manfred}, booktitle = {Berichte der Fachtagung Baustatik – Baupraxis 15, 04. und 05. März 2024, Hamburg}, doi = {10.15480/882.9247}, editor = {Oesterle, Bastian and Bögle, Annette and Weber, Wolfgang and Striefler, Lukas}, pages = {357--364}, title = {Hierarchische Formulierungen für statische und dynamische Analysen von Flächentragwerken}, year = 2024 }
Bieber, S., Oesterle, B., Bischoff, M., & Ramm, E. (2022). Strategy for Preventing Membrane Locking Through Reparametrization. In F. Aldakheel, B. Hudobivnik, M. Soleimani, H. Wessels, C. Weißenfels, & M. Marino (Hrsg.), Current Trends and Open Problems in Computational Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-87312-7_7
- Zusammenfassung
- BibTeX
Zusammenfassung
The contribution takes up the concept of preventing locking a priori in the theory of thin-walled structures instead of curing it during discretization. After briefly summarizing the successful concept for avoiding transverse shear locking through reparametrization of primary variables for beams, plates and shells we concentrate on the same approach for preventing also membrane locking. Here, we describe first steps referring to the plane curved Bernoulli beam as a conceptual proof for the new method. Inspired by the so-called Mixed Displacement variational method we discuss three variants of replacing primary displacement parameters by alternative variables.
BibTeX
@incollection{bieber2022strategy, abstract = {The contribution takes up the concept of preventing locking a priori in the theory of thin-walled structures instead of curing it during discretization. After briefly summarizing the successful concept for avoiding transverse shear locking through reparametrization of primary variables for beams, plates and shells we concentrate on the same approach for preventing also membrane locking. Here, we describe first steps referring to the plane curved Bernoulli beam as a conceptual proof for the new method. Inspired by the so-called Mixed Displacement variational method we discuss three variants of replacing primary displacement parameters by alternative variables.}, author = {Bieber, Simon and Oesterle, Bastian and Bischoff, Manfred and Ramm, Ekkehard}, booktitle = {Current Trends and Open Problems in Computational Mechanics}, doi = {10.1007/978-3-030-87312-7_7}, editor = {Aldakheel, Fadi and Hudobivnik, Blaz and Soleimani, Meisam and Wessels, Henning and Weißenfels, Christian and Marino, Michele}, publisher = {Springer, Cham.}, title = {Strategy for Preventing Membrane Locking Through Reparametrization}, year = 2022 }
Bieber, S., Oesterle, B., Ramm, E., & Bischoff, M. (2018). A variational method to avoid locking – independent of the discretization scheme. International Journal for Numerical Methods in Engineering, 114, 801–827. https://doi.org/10.1002/nme.5766
- Zusammenfassung
- BibTeX
Zusammenfassung
We present a variational method for problems in solid and structural mechanics that is designed to be intrinsically free from locking when using equal order interpolation for all involved fields. The specific feature of the formulation is that it avoids all geometrical locking effects (as opposed to material locking effects, e.g. Poisson locking) for any type of structural or solid model, independent of the underlying discretization scheme. The possibility to employ equal order interpolation for all involved fields circumvents the task of finding particular function spaces to remove locking and avoid artificial stress oscillations. This is particularly attractive for instance for isogeometric analysis using unstructured meshes or T-splines. Comprehensive numerical tests underline the promising behaviour of the proposed method for geometrically linear and non-linear problems in terms of displacements and stress resultants using standard finite elements, isogeometric finite elements and a meshless method.
BibTeX
@article{bieber2018variational, abstract = {We present a variational method for problems in solid and structural mechanics that is designed to be intrinsically free from locking when using equal order interpolation for all involved fields. The specific feature of the formulation is that it avoids all geometrical locking effects (as opposed to material locking effects, e.g. Poisson locking) for any type of structural or solid model, independent of the underlying discretization scheme. The possibility to employ equal order interpolation for all involved fields circumvents the task of finding particular function spaces to remove locking and avoid artificial stress oscillations. This is particularly attractive for instance for isogeometric analysis using unstructured meshes or T-splines. Comprehensive numerical tests underline the promising behaviour of the proposed method for geometrically linear and non-linear problems in terms of displacements and stress resultants using standard finite elements, isogeometric finite elements and a meshless method.}, author = {Bieber, Simon and Oesterle, Bastian and Ramm, Ekkehard and Bischoff, Manfred}, doi = {10.1002/nme.5766}, journal = {International Journal for Numerical Methods in Engineering}, pages = {801-827}, title = {A variational method to avoid locking – independent of the discretization scheme}, volume = { 114}, year = 2018 }
Oesterle, B., Bieber, S., Sachse, R., Ramm, E., & Bischoff, M. (2018). Intrinsically locking-free formulations for isogeometric beam, plate and shell analysis. Proc. Appl. Math. Mech., 18. https://doi.org/10.1002/pamm.201800399
- Zusammenfassung
- BibTeX
Zusammenfassung
In this contribution a class of formulations for beams, plates and shells is presented, which intrinsically avoids locking, independent of the utilized discretization scheme. The key idea is the reparametrization of the kinematic equations to avoid locking on theory level – prior to discretization. Thus, the resulting formulations are locking‐free for any equal‐order interpolation. As demonstrator, we present both mixed and primal concepts for Timoshenko beams in both weak and strong form, as well as their theoretical relationships. Besides a weak form Galerkin‐type solution using B‐Splines, we show the generality of the presented concepts by employing isogeometric collocation based on the corresponding Euler‐Lagrange equations of the boundary value problem. The quality of both stress resultants and displacements is investigated. Although the underlying concept addresses beams, plates and shells, the present contribution illustrates the methodology for the Timoshenko beam.
BibTeX
@inproceedings{oesterle2018intrinsically, abstract = {In this contribution a class of formulations for beams, plates and shells is presented, which intrinsically avoids locking, independent of the utilized discretization scheme. The key idea is the reparametrization of the kinematic equations to avoid locking on theory level – prior to discretization. Thus, the resulting formulations are locking‐free for any equal‐order interpolation. As demonstrator, we present both mixed and primal concepts for Timoshenko beams in both weak and strong form, as well as their theoretical relationships. Besides a weak form Galerkin‐type solution using B‐Splines, we show the generality of the presented concepts by employing isogeometric collocation based on the corresponding Euler‐Lagrange equations of the boundary value problem. The quality of both stress resultants and displacements is investigated. Although the underlying concept addresses beams, plates and shells, the present contribution illustrates the methodology for the Timoshenko beam.}, author = {Oesterle, Bastian and Bieber, Simon and Sachse, Renate and Ramm, Ekkehard and Bischoff, Manfred}, booktitle = {Proc. Appl. Math. Mech., 18}, doi = {10.1002/pamm.201800399}, title = {Intrinsically locking-free formulations for isogeometric beam, plate and shell analysis}, year = 2018 }
Oesterle, B., Sachse, R., Ramm, E., & Bischoff, M. (2017). Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization. Computer Methods in Applied Mechanics and Engineering, 321, 383–405. https://doi.org/10.1016/j.cma.2017.03.031
- Zusammenfassung
- BibTeX
Zusammenfassung
Two novel hierarchic finite element formulations for geometrically nonlinear shell analysis including the effects of transverse shear are presented. Both methods combine a fully nonlinear Kirchhoff-Love shell model with hierarchically added linearized transverse shear components. Thus, large rotations can be taken into account while circumventing the peculiar task of finding a corresponding parametrization of the rotation tensor. The two formulations differ in the way the transverse shear effects are included, either using hierarchic rotations or hierarchic displacements. The underlying assertion is that in most practical applications the transverse shear angles are small even for large deformations. This is confirmed by various numerical experiments. The hierarchic construction results in an additive strain decomposition into parts resulting from membrane and bending deformation and additional contributions from transverse shear. It requires at least C1-continuous shape functions, which can be easily established within the isogeometric context using spline based finite elements. As reported earlier, this concept is intrinsically free from transverse shear locking. In the nonlinear case it dramatically facilitates representation of large rotations in shell analysis.
BibTeX
@article{oesterle2017hierarchic, abstract = {Two novel hierarchic finite element formulations for geometrically nonlinear shell analysis including the effects of transverse shear are presented. Both methods combine a fully nonlinear Kirchhoff-Love shell model with hierarchically added linearized transverse shear components. Thus, large rotations can be taken into account while circumventing the peculiar task of finding a corresponding parametrization of the rotation tensor. The two formulations differ in the way the transverse shear effects are included, either using hierarchic rotations or hierarchic displacements. The underlying assertion is that in most practical applications the transverse shear angles are small even for large deformations. This is confirmed by various numerical experiments. The hierarchic construction results in an additive strain decomposition into parts resulting from membrane and bending deformation and additional contributions from transverse shear. It requires at least C1-continuous shape functions, which can be easily established within the isogeometric context using spline based finite elements. As reported earlier, this concept is intrinsically free from transverse shear locking. In the nonlinear case it dramatically facilitates representation of large rotations in shell analysis.}, author = {Oesterle, Bastian and Sachse, Renate and Ramm, Ekkehard and Bischoff, Manfred}, doi = {10.1016/j.cma.2017.03.031}, journal = {Computer Methods in Applied Mechanics and Engineering}, pages = {383–405}, title = {Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization}, volume = { 321}, year = 2017 }
Oesterle, B., Ramm, E., & Bischoff, M. (2016). A shear deformable, rotation-free isogeometric shell formulation. Computer Methods in Applied Mechanics and Engineering, 307, 235–255. https://doi.org/10.1016/j.cma.2016.04.015
- Zusammenfassung
- BibTeX
Zusammenfassung
A finite element formulation for a geometrically linear, shear deformable (Reissner–Mindlin type) shell theory is presented, which exclusively uses displacement degrees of freedom. The total displacement is subdivided into a part representing the membrane and bending deformation, enriched by two extra “shear displacements”, representing transverse shear deformation. This rotation-free approach is accomplished within the isogeometric concept, using C1-continuous, quadratic NURBS as shape functions. The particular displacement parametrization decouples transverse shear from bending and thus the formulation is free from transverse shear locking by construction, i.e. locking is avoided on the theory level, not by choice of a particular discretization. Compared to the hierarchic formulation proposed earlier within the group of the authors (Echter et al., 2013), the method presented herein avoids artificial oscillations of the transverse shear forces. Up to now, a similar, displacement based method to avoid membrane locking has not been found. Thus, in the present formulation the mixed method from Echter et al. (2013) is used to avoid membrane locking.
BibTeX
@article{oesterle2016shear, abstract = {A finite element formulation for a geometrically linear, shear deformable (Reissner–Mindlin type) shell theory is presented, which exclusively uses displacement degrees of freedom. The total displacement is subdivided into a part representing the membrane and bending deformation, enriched by two extra “shear displacements”, representing transverse shear deformation. This rotation-free approach is accomplished within the isogeometric concept, using C1-continuous, quadratic NURBS as shape functions. The particular displacement parametrization decouples transverse shear from bending and thus the formulation is free from transverse shear locking by construction, i.e. locking is avoided on the theory level, not by choice of a particular discretization. Compared to the hierarchic formulation proposed earlier within the group of the authors (Echter et al., 2013), the method presented herein avoids artificial oscillations of the transverse shear forces. Up to now, a similar, displacement based method to avoid membrane locking has not been found. Thus, in the present formulation the mixed method from Echter et al. (2013) is used to avoid membrane locking.}, author = {Oesterle, Bastian and Ramm, Ekkehard and Bischoff, Manfred}, doi = {10.1016/j.cma.2016.04.015}, journal = {Computer Methods in Applied Mechanics and Engineering}, pages = {235–255}, title = {A shear deformable, rotation-free isogeometric shell formulation}, volume = { 307}, year = 2016 }
Echter, R., Oesterle, B., & Bischoff, M. (2013). A hierarchic family of isogeometric shell finite elements. Computer Methods in Applied Mechanics and Engineering, 254, 170–180. https://doi.org/10.1016/j.cma.2012.10.018
- Zusammenfassung
- BibTeX
Zusammenfassung
A hierarchic family of isogeometric shell finite elements based on NURBS shape functions is presented. In contrast to classical shell finite element formulations, inter-element continuity of at least C1 enables a unique and continuous representation of the surface normal within one NURBS patch. This does not only facilitate formulation of Kirchhoff-Love type shell models, for which the standard Galerkin weak form has a variational index of 2, but it also offers significant advantages for shear deformable (Reissner-Mindlin type) shells and higher order shell models. For a 5-parameter shell formulation with Reissner-Mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated Kirchhoff-Love type director of the deformed configuration. This split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration, assumed natural strains or mixed finite element formulations. The third member of the hierarchy is a 7-parameter model including thickness change and allowing for application of unmodified three-dimensional constitutive laws. The phenomenon of curvature thickness locking, coming along with this kinematic extension, again is automatically avoided by the hierarchic difference vector concept without any further treatment. Membrane locking and in-plane shear locking are removed by two different approaches: firstly elimination via the Discrete Strain Gap (DSG) method and secondly removal of parasitic membrane strains using a hybrid-mixed method based on the Hellinger-Reissner variational principle. The hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach.
BibTeX
@article{echter2013hierarchic, abstract = {A hierarchic family of isogeometric shell finite elements based on NURBS shape functions is presented. In contrast to classical shell finite element formulations, inter-element continuity of at least C1 enables a unique and continuous representation of the surface normal within one NURBS patch. This does not only facilitate formulation of Kirchhoff-Love type shell models, for which the standard Galerkin weak form has a variational index of 2, but it also offers significant advantages for shear deformable (Reissner-Mindlin type) shells and higher order shell models. For a 5-parameter shell formulation with Reissner-Mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated Kirchhoff-Love type director of the deformed configuration. This split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration, assumed natural strains or mixed finite element formulations. The third member of the hierarchy is a 7-parameter model including thickness change and allowing for application of unmodified three-dimensional constitutive laws. The phenomenon of curvature thickness locking, coming along with this kinematic extension, again is automatically avoided by the hierarchic difference vector concept without any further treatment. Membrane locking and in-plane shear locking are removed by two different approaches: firstly elimination via the Discrete Strain Gap (DSG) method and secondly removal of parasitic membrane strains using a hybrid-mixed method based on the Hellinger-Reissner variational principle. The hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach.}, author = {Echter, R. and Oesterle, B. and Bischoff, M.}, doi = {10.1016/j.cma.2012.10.018}, journal = {Computer Methods in Applied Mechanics and Engineering}, pages = {170-180}, title = {A hierarchic family of isogeometric shell finite elements}, volume = { 254}, year = 2013 }

Bearbeitung:

Dieses Bild zeigt Tarun Kumar Mitruka Vinod Kumar Mitruka

Intrinsisch lockingfreie Formulierungen

Überblick

Projektbeschreibung

Hintergrund

Hierarchische Formulierungen und eine Mixed-Displacement-Formulierung

Herausforderungen und Ziel der Forschung

Projektdaten

Veröffentlichungen

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Bearbeitung:

Tarun Kumar Mitruka Vinod Kumar Mitruka

Zielgruppe

Formalia

Services

Organisation

Intrinsisch lockingfreie Formulierungen

Überblick

Projektbeschreibung

Hintergrund

Hierarchische Formulierungen und eine Mixed-Displacement-Formulierung

Herausforderungen und Ziel der Forschung

Projektdaten

Veröffentlichungen

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Zusammenfassung

BibTeX

Bearbeitung:

Tarun Kumar Mitruka Vinod Kumar Mitruka

So erreichen Sie uns

Zielgruppe

Formalia

Services

Organisation