A primal-dual active set algorithm for three-dimensional contact problems with coulomb friction

International audienceIn this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth com-plementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples

Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

This is the peer reviewed version of the following article: [Agelet de Saracibar C, Di Capua D. Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method. Int J Numer Methods Eng. 2018;113:910–937. https://doi.org/10.1002/nme.5693], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5693. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.In this paper, conserving time-stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed in order to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy-momentum preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty-based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time stepping algorithm is shown.Peer ReviewedPostprint (author's final draft

Assessment of mechanical properties of human head tissues for trauma modelling

[EN] Many discrepancies are found in the literature regarding the damage and constitutive models for head tissues as well as the values of the constants involved in the constitutive equations. Their proper definition is required for consistent numerical model performance when predicting human head behaviour, and hence skull fracture and brain damage. The objective of this research is to perform a critical review of constitutive models and damage indicators describing human head tissue response under impact loading. A 3D finite element human head model has been generated by using computed tomography images, which has been validated through the comparison to experimental data in the literature. The threshold values of the skull and the scalp that lead to fracture have been analysed. We conclude that (1) compact bone properties are critical in skull fracture, (2) the elastic constants of the cerebrospinal fluid affect the intracranial pressure distribution, and (3) the consideration of brain tissue as a nearly incompressible solid with a high (but not complete) water content offers pressure responses consistent with the experimental data.Generalitat Valenciana, Grant/Award Number: PROMETEO 2016/007; Ministerio de Economia y Compatitividad and Fondo Europeo de Desarrollo Regional, Grant/Award Number: RTC-2015-3887-8Lozano-Mínguez, E.; Palomar-Toledano, M.; Infante, D.; Rupérez Moreno, MJ.; Giner Maravilla, E. (2018). Assessment of mechanical properties of human head tissues for trauma modelling. International Journal for Numerical Methods in Biomedical Engineering. 34(5):1-17. https://doi.org/10.1002/cnm.2962S117345Hyder, A. A., Wunderlich, C. A., Puvanachandra, P., Gururaj, G., & Kobusingye, O. C. (2007). The impact of traumatic brain injuries: A global perspective. NeuroRehabilitation, 22(5), 341-353. doi:10.3233/nre-2007-22502Meaney, D. F., Morrison, B., & Dale Bass, C. (2014). The Mechanics of Traumatic Brain Injury: A Review of What We Know and What We Need to Know for Reducing Its Societal Burden. Journal of Biomechanical Engineering, 136(2). doi:10.1115/1.4026364Report Violence and Injury Prevention and Disability (VIP)-neurotrauma 2010 http://www.who.int/violence_injury_prevention/road_traffic/activities/neurotrauma/en/Deng, X., Potula, S., Grewal, H., Solanki, K. N., Tschopp, M. A., & Horstemeyer, M. F. (2013). Finite element analysis of occupant head injuries: Parametric effects of the side curtain airbag deployment interaction with a dummy head in a side impact crash. Accident Analysis & Prevention, 55, 232-241. doi:10.1016/j.aap.2013.03.016Marjoux, D., Baumgartner, D., Deck, C., & Willinger, R. (2008). Head injury prediction capability of the HIC, HIP, SIMon and ULP criteria. Accident Analysis & Prevention, 40(3), 1135-1148. doi:10.1016/j.aap.2007.12.006Bolander, R., Mathie, B., Bir, C., Ritzel, D., & VandeVord, P. (2011). Skull Flexure as a Contributing Factor in the Mechanism of Injury in the Rat when Exposed to a Shock Wave. Annals of Biomedical Engineering, 39(10), 2550-2559. doi:10.1007/s10439-011-0343-0Li, G., Zhang, J., Wang, K., Wang, M., Gao, C., & Ma, C. (2016). Experimental research of mechanical behavior of porcine brain tissue under rotational shear stress. Journal of the Mechanical Behavior of Biomedical Materials, 57, 224-234. doi:10.1016/j.jmbbm.2015.12.002YOGANANDAN, N., PINTAR, F. A., SANCES, A., WALSH, P. R., EWING, C. L., THOMAS, D. J., & SNYDER, R. G. (1995). Biomechanics of Skull Fracture. Journal of Neurotrauma, 12(4), 659-668. doi:10.1089/neu.1995.12.659Motherway, J. A., Verschueren, P., Van der Perre, G., Vander Sloten, J., & Gilchrist, M. D. (2009). The mechanical properties of cranial bone: The effect of loading rate and cranial sampling position. Journal of Biomechanics, 42(13), 2129-2135. doi:10.1016/j.jbiomech.2009.05.030Lakatos, É., Magyar, L., & Bojtár, I. (2014). Material Properties of the Mandibular Trabecular Bone. Journal of Medical Engineering, 2014, 1-7. doi:10.1155/2014/470539Boruah, S., Subit, D. L., Paskoff, G. R., Shender, B. S., Crandall, J. R., & Salzar, R. S. (2017). Influence of bone microstructure on the mechanical properties of skull cortical bone – A combined experimental and computational approach. Journal of the Mechanical Behavior of Biomedical Materials, 65, 688-704. doi:10.1016/j.jmbbm.2016.09.041McElhaney, J. H., Fogle, J. L., Melvin, J. W., Haynes, R. R., Roberts, V. L., & Alem, N. M. (1970). Mechanical properties of cranial bone. Journal of Biomechanics, 3(5), 495-511. doi:10.1016/0021-9290(70)90059-xKleiven, S. (2003). Influence of Impact Direction on the Human Head in Prediction of Subdural Hematoma. Journal of Neurotrauma, 20(4), 365-379. doi:10.1089/089771503765172327Miller, L. E., Urban, J. E., & Stitzel, J. D. (2016). Development and validation of an atlas-based finite element brain model. Biomechanics and Modeling in Mechanobiology, 15(5), 1201-1214. doi:10.1007/s10237-015-0754-1Tse, K. M., Tan, L. B., Lee, S. J., Lim, S. P., & Lee, H. P. (2015). Investigation of the relationship between facial injuries and traumatic brain injuries using a realistic subject-specific finite element head model. Accident Analysis & Prevention, 79, 13-32. doi:10.1016/j.aap.2015.03.012Kimpara, H., Nakahira, Y., Iwamoto, M., Miki, K., Ichihara, K., Kawano, S., & Taguchi, T. (2006). Investigation of Anteroposterior Head-Neck Responses during Severe Frontal Impacts Using a Brain-Spinal Cord Complex FE Model. SAE Technical Paper Series. doi:10.4271/2006-22-0019Brands, D. W. A., Bovendeerd, P. H. M., & Wismans, J. S. H. M. (2002). On the Potential Importance of Non-Linear Viscoelastic Material Modelling for Numerical Prediction of Brain Tissue Response: Test and Application. SAE Technical Paper Series. doi:10.4271/2002-22-0006Kleiven, S. (2007). Predictors for Traumatic Brain Injuries Evaluated through Accident Reconstructions. SAE Technical Paper Series. doi:10.4271/2007-22-0003Ho, J., Zhou, Z., Li, X., & Kleiven, S. (2017). The peculiar properties of the falx and tentorium in brain injury biomechanics. Journal of Biomechanics, 60, 243-247. doi:10.1016/j.jbiomech.2017.06.023Dassault Systèmes Abaqus 6.12 User's Manual 2012Systèmes D Abaqus 6.12 Analysis User's Manual 2012Tadepalli, S. C., Erdemir, A., & Cavanagh, P. R. (2011). Comparison of hexahedral and tetrahedral elements in finite element analysis of the foot and footwear. Journal of Biomechanics, 44(12), 2337-2343. doi:10.1016/j.jbiomech.2011.05.006Hüeber, S., Mair, M., & Wohlmuth, B. I. (2005). A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems. Applied Numerical Mathematics, 54(3-4), 555-576. doi:10.1016/j.apnum.2004.09.019Kleiven, S., & von Holst, H. (2002). Consequences of head size following trauma to the human head. Journal of Biomechanics, 35(2), 153-160. doi:10.1016/s0021-9290(01)00202-0Zhou C Khalil TB King AI A new model comparing impact responses of the homogeneous and inhomogeneous human brain 1995WILLINGER, R., TALEB, L., & KOPP, C.-M. (1995). Modal and Temporal Analysis of Head Mathematical Models. Journal of Neurotrauma, 12(4), 743-754. doi:10.1089/neu.1995.12.743Ruan JS Khalil TB King AI Finite element modeling of direct head impact 1993 https://doi.org/10.4271/933114Tan, L. B., Chew, F. S., Tse, K. M., Chye Tan, V. B., & Lee, H. P. (2014). Impact of complex blast waves on the human head: a computational study. International Journal for Numerical Methods in Biomedical Engineering, 30(12), 1476-1505. doi:10.1002/cnm.2668Nahum AM Smith R Ward CC Intracranial pressure dynamics during head impact 1977Gilchrist, M. D., & O’Donoghue, D. (2000). Simulation of the development of frontal head impact injury. Computational Mechanics, 26(3), 229-235. doi:10.1007/s004660000179Willinger, R., Kang, H.-S., & Diaw, B. (1999). Three-Dimensional Human Head Finite-Element Model Validation Against Two Experimental Impacts. Annals of Biomedical Engineering, 27(3), 403-410. doi:10.1114/1.165Yang, B., Tse, K.-M., Chen, N., Tan, L.-B., Zheng, Q.-Q., Yang, H.-M., … Lee, H.-P. (2014). Development of a Finite Element Head Model for the Study of Impact Head Injury. BioMed Research International, 2014, 1-14. doi:10.1155/2014/408278Mihai, L. A., Budday, S., Holzapfel, G. A., Kuhl, E., & Goriely, A. (2017). A family of hyperelastic models for human brain tissue. Journal of the Mechanics and Physics of Solids, 106, 60-79. doi:10.1016/j.jmps.2017.05.015Han, I. S., & Kim, Y. E. (2014). Development of a new head/brain model for the prediction of subdural hemorrhage. International Journal of Precision Engineering and Manufacturing, 15(11), 2405-2411. doi:10.1007/s12541-014-0607-3Moran, R., Smith, J. H., & García, J. J. (2014). Fitted hyperelastic parameters for Human brain tissue from reported tension, compression, and shear tests. Journal of Biomechanics, 47(15), 3762-3766. doi:10.1016/j.jbiomech.2014.09.030Mendis, K. K., Stalnaker, R. L., & Advani, S. H. (1995). A Constitutive Relationship for Large Deformation Finite Element Modeling of Brain Tissue. Journal of Biomechanical Engineering, 117(3), 279-285. doi:10.1115/1.2794182Sahoo, D., Deck, C., & Willinger, R. (2014). Development and validation of an advanced anisotropic visco-hyperelastic human brain FE model. Journal of the Mechanical Behavior of Biomedical Materials, 33, 24-42. doi:10.1016/j.jmbbm.2013.08.022Belingardi G Chiandussi G Gaviglio I Development and validation of a new finite element model of human head Proceedings of 19th International Technical Conference on the Enhanced Safety of Vehicles 2005Tse, K. M., Tan, L. B., Lee, S. J., Lim, S. P., & Lee, H. P. (2013). Development and validation of two subject-specific finite element models of human head against three cadaveric experiments. International Journal for Numerical Methods in Biomedical Engineering, 30(3), 397-415. doi:10.1002/cnm.2609Yang, J. (2011). Investigation of Brain Trauma Biomechanics in Vehicle Traffic Accidents Using Human Body Computational Models. Computational Biomechanics for Medicine, 5-14. doi:10.1007/978-1-4419-9619-0_2Mao, H., Gao, H., Cao, L., Genthikatti, V. V., & Yang, K. H. (2013). Development of high-quality hexahedral human brain meshes using feature-based multi-block approach. Computer Methods in Biomechanics and Biomedical Engineering, 16(3), 271-279. doi:10.1080/10255842.2011.617005Yan, W., & Pangestu, O. D. (2011). A modified human head model for the study of impact head injury. Computer Methods in Biomechanics and Biomedical Engineering, 14(12), 1049-1057. doi:10.1080/10255842.2010.506435Baeck K Goffin J Vander Sloten J The use of different CSF representations in a numerical head model and their effect on the results of FE head impact analyses 2011 http://www.dynalook.com/8th-european-ls-dyna-conference/session-7/Session7_Paper3.pdfKleiven, S. (2006). Evaluation of head injury criteria using a finite element model validated against experiments on localized brain motion, intracerebral acceleration, and intracranial pressure. International Journal of Crashworthiness, 11(1), 65-79. doi:10.1533/ijcr.2005.0384Galford, J. E., & McElhaney, J. H. (1970). A viscoelastic study of scalp, brain, and dura. Journal of Biomechanics, 3(2), 211-221. doi:10.1016/0021-9290(70)90007-2Bradshaw D Morfey C Pressure and shear response in brain injury models Proceedings of the 17th International Technical Conference on the Enhanced Safety of Vehicles 2001 1 10Mooney, M. (1940). A Theory of Large Elastic Deformation. Journal of Applied Physics, 11(9), 582-592. doi:10.1063/1.1712836Large elastic deformations of isotropic materials IV. further developments of the general theory. (1948). Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), 379-397. doi:10.1098/rsta.1948.0024Melvin JW McElhaney JH Roberts VL Development of a mechanical model of the human head-determination of tissue properties and synthetic substitute materials 1970Raul, J.-S., Baumgartner, D., Willinger, R., & Ludes, B. (2005). Finite element modelling of human head injuries caused by a fall. International Journal of Legal Medicine, 120(4), 212-218. doi:10.1007/s00414-005-0018-1Sahoo, D., Deck, C., Yoganandan, N., & Willinger, R. (2016). Development of skull fracture criterion based on real-world head trauma simulations using finite element head model. Journal of the Mechanical Behavior of Biomedical Materials, 57, 24-41. doi:10.1016/j.jmbbm.2015.11.014Silver, F. H. (1994). Biomaterials, Medical Devices and Tissue Engineering: An Integrated Approach. doi:10.1007/978-94-011-0735-8Dassault Systèmes Section 1.2.19 VUSDFLD, Abaqus User Subroutines Reference Manual 2012Hambli, R. (2012). A quasi-brittle continuum damage finite element model of the human proximal femur based on element deletion. Medical & Biological Engineering & Computing, 51(1-2), 219-231. doi:10.1007/s11517-012-0986-5Harrison, N. M., McDonnell, P., Mullins, L., Wilson, N., O’Mahoney, D., & McHugh, P. E. (2012). Failure modelling of trabecular bone using a non-linear combined damage and fracture voxel finite element approach. Biomechanics and Modeling in Mechanobiology, 12(2), 225-241. doi:10.1007/s10237-012-0394-7Vavalle, N. A., Davis, M. L., Stitzel, J. D., & Gayzik, F. S. (2015). Quantitative Validation of a Human Body Finite Element Model Using Rigid Body Impacts. Annals of Biomedical Engineering, 43(9), 2163-2174. doi:10.1007/s10439-015-1286-7Delye, H., Verschueren, P., Depreitere, B., Van Lierde, C., Verpoest, I., Berckmans, D., … Goffin, J. (2005). Biomechanics of Frontal Skull Fracture. Solid Mechanics and Its Applications, 185-193. doi:10.1007/1-4020-3796-1_19Ruan, J. S., Khalil, T., & King, A. I. (1994). Dynamic Response of the Human Head to Impact by Three-Dimensional Finite Element Analysis. Journal of Biomechanical Engineering, 116(1), 44-50. doi:10.1115/1.2895703Chafi, M. S., Dirisala, V., Karami, G., & Ziejewski, M. (2009). A finite element method parametric study of the dynamic response of the human brain with different cerebrospinal fluid constitutive properties. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 223(8), 1003-1019. doi:10.1243/09544119jeim631Ruan, J., & Prasad, P. (2006). The influence of human head tissue properties on intracranial pressure response during direct head impact. International Journal of Vehicle Safety, 1(4), 281. doi:10.1504/ijvs.2006.01123

Diskretisierungstechniken und effiziente Algorithmen für Kontaktprobleme

This thesis is concerned with the development of efficient numerical solution algorithms for nonlinear contact problems with friction. Such type of problems play an important role in many technical and engineering applications. Thus, the design of discretization techniques and efficient solution strategies is still a challenging task both from the engineering and the mathematical point of view. Domain decomposition techniques based on finite element methods are a powerful tool to approximate the solution of partial differential equations as they occur in the framework of structural mechanics. Here, we focus on discretization techniques based on the mortar method by introducing an additional unknown named Lagrange multiplier or dual variable in order to formulate the interface constraints between the involved bodies. In the framework of contact problems, where the weak formulation consists of a variational inequality, this additional variable models the contact stresses at the common contact interface. Using standard finite elements for the discretization of the Lagrange multiplier, the contact conditions result in a segment-to-segment approach, where the mechanical inequality constraints can only be resolved by some global optimization procedure on the contact boundary. This can be avoided by working with locally defined dual or biorthogonal basis functions for the Lagrange multiplier space. Then, the segment-to-segment approach is algebraically equivalent to a node-to-segment approach, and the inequality constraints decouple point-wise. Additionally, we are able to transform a two-body contact problem into a one-body problem by a local preprocess, and hence apply the same nonlinear solver. Mathematically, the preprocess is equivalent to a basis transformation; physically, master and slave side are glued together such that the two bodies form a composite material and the displacement on the slave side reflects the relative displacement between the two bodies. In this thesis, we analyze the discretization error of the proposed mortar formulation and give optimal a priori error estimates. A various set of numerical examples are given to confirm the achieved theoretical results. The decoupled contact constraints provide a basis for the construction of efficient solution algorithms. The presented numerical approaches are semi-smooth Newton methods which are equivalent to a primal-dual active set strategy in the case without friction. The point-wise inequality constraints between the primal variable, i.e., the displacement, and the dual variable, i.e., the Lagrange multiplier, are written as an equality constraint by the use of a semi-smooth nonlinear complementarity function. Even for the case of contact problems including friction with Coulomb's friction law we are able to construct a full semi-smooth Newton algorithm. Due to the use of the dual basis functions for the Lagrange multiplier, we are able to locally eliminate the degrees of freedom for the dual variable. Thus, in each iteration step, we have to solve a linear system with respect to the primal variable, where, the contact constraints enter as boundary conditions of Dirichlet-, Neumann-, or Robin-type. Therefore, existing finite elements codes for structural mechanics can be easily extended to the case of contact problems by using the proposed methods. Using iterative solvers like optimal multigrid methods to solve the arising linear system in each step, we are able to construct inexact strategies, where the linear system is not completely solved in each Newton step. By this, we get an efficient algorithm for solving a fully nonlinear contact problem whose additional cost is negligible compared to solving a linear system. Several numerical examples are provided to investigate the performance and efficiency of the introduced algorithms. In the last part of this thesis, we extend the proposed formulation and the efficient solution algorithms to more general applications. Firstly, we adapt our solution strategies to the case of dynamical contact problems in combination with nonlinear material laws. Especially, we focus on energy-conserving algorithms. Secondly, we treat thermo-mechanical contact problems, where, the temperature is introduced as an additional unknown. This extension is quite natural since heat is generated due to the mechanical frictional work. Similar as the mechanical Lagrange multiplier takes care on the mechanical contact constraints, a thermal Lagrange multiplier modeling the heat flux across the contact interface is added to enforce the thermal flux conditions over the contact interface. We propose a mortar formulation for this Robin-type thermal interface conditions and extend our contact algorithms to solve the resulting nonlinear problem.Die numerische Simulation von Vorgngen in Natur und Technik nimmt in heutiger Zeit eine stetig zunehmende Rolle in der Entwicklung neuer Produkte in der Industrie aber auch im Verständnis von Naturphänomenen ein. Selbst in der Medizin werden mittlerweile numerische Simulationsmethoden beispielsweise zur Planung von komplizierten Operationen eingesetzt. Mathematisch werden alle diese Vorgänge mit Hilfe von partiellen Differentialgleichungen beschrieben, deren analytische Lösungen im Allgemeinen nicht bekannt sind. Geeignete Diskretisierungsmethoden, wie beispielsweise Finite Element Methoden in Kombination mit effizienten Lösungsverfahren, stellen daher ein mächtiges Werkzeug dar, um näherungsweise Lösungen dieser Systeme zu erhalten. Diese Arbeit befasst sich mit der Diskretisierung und der Entwicklung effizienter Lösungsverfahren für Kontaktprobleme in der nichtlinearen Strukturmechanik. Kontaktprobleme spielen nicht nur im Alltag, sondern auch in vielen ingenieurswissenschaftlichen Fragestellungen ein große Rolle. Beispielsweise stellt jede Art der Fortbewegung, sei es das Gehen oder der rollende Reifen eines Autos einen reibungsbehafteten Kontaktvorgang dar. In der Technik findet man derartige Vorgänge zum Beispiel in der Blechumformung oder beim Crash zweier Fahrzeuge. Gerade hier ist man an einer effizienten Simulation bereits im Entwicklungsstadium eines Fahrzeuges sehr interessiert. Da bei der Simulation solcher Vorgänge meist mehrere Objekte beteiligt sind und diskretisiert werden müssen, stellen moderne Gebietszerlegungsmethoden einen hervorragenden Ausgangspunkt für die Entwicklung effizienter Lösungsalgorithmen dar. Insbesondere eignet sich hierfrü die Mortar-Methode, welche einen mathematisch analysierbaren Zugang liefert, der ein nichtkonformes Vernetzen der beteiligten Objekte an ihren Schnittstellen, den Kontaktflächen, erlaubt. Somit können die Rechengitter den Eigenschaften der einzelnen Objekte separat erzeugt und deren Gestalt optimal angepasst werden. Der Informationsaustausch zwischen den einzelnen Teilgebeiten geschieht bei der Mortar-Methode mit Hilfe einer zusätzlichen Variablen, dem Lagrange-Multiplikator. Im Falle eines Kontaktvorgangs berschreibt dieser gerade die Kontaktkräfte, die zwischen den beteiligten Objekten ausgetauscht werden. Für die diskrete Beschreibung des Lagrange-Multiplikators werden in dieser Arbeit duale Basisfunktionen verwendet. Die Verwendung dieser Basisfunktionen führt zu einer Entkopplung der Kontaktnebenbedingungen an den einzelnen Diskretisierungspunkten an der Kontaktfläche, welche dann einen hervorragenden Ausgangspunkt für die Entwicklung effizienter iterativer Lösungsalgorithmen für das entstehende nichtlineare Gleichungssystem basierend auf nichtglatten Newton-Verfahren darstellt. Somit ist der Zugang mit Mortar-Techniken, der in seiner Grundform eine Segment-to-Segment-Formulierung darstellt, in seiner algebraischen Form äquivalent zu einer Node-to-Segment-Formulierung. Weiter lassen sich aufgrund der Wahl dieser speziellen Basisfunktionen im resultierenden algebraischen Gleichungssystem die zusätzlichen Freiheitsgade für den Lagrange-Multiplikator auf effiziente Weise lokal eliminieren, so dass am Ende nur noch ein auskondensiertes System bezüglich der primalen Variable gelöst werden muss. Nichtglatte Newton-Verfahren gewinnen in modernen Simulationstechniken stetig an Bedeutung. Im Rahmen dieser Arbeit soll ihre Anwendung auf Kontaktprobleme in Kombination mit der Mortar-Methode vorgestellt und untersucht werden

A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction

In this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth complementarity function for the 3D friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization is applied, and local superlinear convergence can be observed. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples.Deutsche Forschungsgemeinschaft, SFB 404, B8, SPP 114

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb‐type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.publishedVersio