Nils Zander, M.Sc. Research Assistant Email: Diese E-Mail-Adresse ist vor Spambots geschützt! Zur Anzeige muss JavaScript eingeschaltet sein!

Research

• Simulation in applied mechanics
• Electro-thermo-mechanical modeling of Field Assisted Sintering Technology using high-order finite elements
• High-order Finite Elements
• Adaptive discretization methods

Publications

Submitted preprints



Journal Papers (peer-reviewed)

• Kollmannsberger, S.; Özcan, A.; Carraturo, M.; Zander, N.; Rank, E. Show abstract A hierarchical computational model for moving thermal loads and phase changes with applications to Selective Laser Melting
  Computers & Mathematics with Applications 75 (5), pp. 1483-1497, 2018
  DOI: 10.1016/j.camwa.2017.11.014 - Status: Verlagsversion / published
  
  
  
  Computational heat transfer analysis often involves moving fluxes which induce traveling fronts of phase change coupled to one or more field variables. Examples are the transient simulation of melting, welding or of additive manufacturing processes, where material changes its state and the controlling fields are temperature and structural deformation. One of the challenges for a numerical computation of these processes is their multi-scale nature with a highly localized zone of phase transition which may travel over a large domain of a body. Here, a transient local adaptation of the approximation, with not only a refinement at the phase front, but also a de-refinement in regions, where the front has past is of advantage because the de-refinement can assure a bounded number of degrees of freedom which is independent from the traveling length of the front. We present a computational model of this process which involves three novelties: a) a very low number of degrees of freedom which yet yields a comparatively high accuracy. The number of degrees of freedom is, additionally, kept practically constant throughout the duration of the simulation. This is achieved by means of the multi-level hp-finite element method. Its exponential convergence is verified for the first time against a semi-analytic, three-dimensional transient linear thermal benchmark with a traveling source term which models a laser beam. b) A hierarchical treatment of the state variables. To this end, the state of the material is managed on a separate, octree-like grid. This material grid may refine or coarsen independently of the discretization used for the temperature field. This methodology is verified against an analytic benchmark of a melting bar computed in three dimensions in which phase changes of the material occur on a rapidly advancing front. c) The combination of these technologies to demonstrate its potential for the computational modeling of selective laser melting processes. To this end, the computational methodology is extended by the finite cell method which allows for accurate simulations in an embedded domain setting. This opens the new modeling possibility that neither a scan vectors no a layer of material needs to conform to the discretization of the finite element mesh but can form only a fraction within the discretization of the field and state variables.
  
  
  
• Elhaddad, M.; Zander, N.; Bog, T.; Kudela, L.; Kollmannsberger, S.; Kirschke, J.S.; Baum, T.; Ruess, M.; Rank, E. Multi-level hp-finite cell method for embedded interface problems with application in biomechanics
  International Journal for Numerical Methods in Biomedical Engineering 34 (4), pp. e2951, Wiley, 2017
  DOI: 10.1002/cnm.2951 - Status: Verlagsversion / published
  
  
  
  
  
  
  
• Bog, T.; Zander, N.; Kollmannsberger, S.; Rank, E. Show abstract Weak imposition of frictionless contact constraints on automatically recovered high-order, embedded interfaces using the finite cell method
  Computational Mechanics, 2017 - Status: Postprint / reviewed
  
  
  
  The finite cell method (FCM) is a fictitious domain approach that greatly simplifies simulations involving complex structures. Recently, the FCM has been applied to contact problems. The current study continues in this field by extending the concept of weakly enforced boundary conditions to inequality constraints for frictionless contact. Furthermore, it formalizes an approach that automatically recovers highorder contact surfaces of (implicitly defined) embedded geometries by means of an extended Marching Cubes (MC) algorithm. To further improve the accuracy of the discretization, irregularities at the boundary of contact zones are treated with multi-level hp-refinements. Numerical results and a systematic study of h-, p- and hprefinements show that the FCM can efficiently provide accurate results for problems involving contact.
  
  
  
• Jomo, J.; Zander, N.; Elhaddad, M.; Özcan, A.; Kollmannsberger, S.; Mundani, R-P.; Rank, E. Parallelization of the multi-level hp-adaptive finite cell method
  Computers and Mathematics with Applications 74 (1), pp. 126-142, Elsevier, 2017
  DOI: 10.1016/j.camwa.2017.01.004 - Status: Verlagsversion / published
  
  
  
  
  
  
  
• D'Angella, D.; Zander, N.; Kollmannsberger, S.; Frischmann, F.; Rank, E.; Schröder, A.; Reali, A. Show abstract Multi-Level hp-Adaptivity and Explicit Error Estimation
  Advanced Modeling and Simulation in Engineering Sciences, pp. 1-18, Springer, 2016
  DOI: 10.1186/s40323-016-0085-5 - Status: Verlagsversion / published
  
  
  
  Recently, a multi-level hp-version of the Finite Element Method (FEM) was proposed to ease the difficulties of treating hanging nodes, while providing full hp-approximation capabilities. In the original paper, the refinement procedure made use of a-priori knowledge of the solution. However, adaptive procedures can produce discretizations which are more effective than an intuitive choice of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable. The present contribution demonstrates that multi-level hp-adaptive schemes can be efficiently driven by an explicit a-posteriori error estimator. To this end, we adopt the classical residual-based error estimator. The main insight here is that its extension to multi-level hp-FEM is possible by considering the refined-most overlay elements as integration domains. We demonstrate on several two- and three-dimensional examples that exponential convergence rates can be obtained.
  
  
  
• Di Stolfo, P.; Schröder, A.; Zander, N.; Kollmannsberger, S. Show abstract An easy treatment of hanging nodes in hp-finite elements
  Finite Elements in Analysis and Design 121, pp. 101-117, 2016
  DOI: 10.1016/j.finel.2016.07.001 - Status: Verlagsversion / published
  
  
  
  We present an easy treatment of (multi-level) hanging nodes in hp-finite elements in two dimensions. Its simplicity is due to the fact that the connectivity matrices can directly be computed in a row-wise fashion. This is achieved by treating a global degree of freedom as a local degree of freedom on each element in an appropriate union of elements. This forms a type of support which is larger than the one used in the conventional approach and generalizes the recently presented multi-level hp-method. Numerical experiments compare the new approach with respect to accuracy and the conditioning of the resulting linear system. We conclude that this new approach achieves nearly the same properties as the conventional approach in all investigated examples and provides an excellent trade-off between implementational complexity and solvability.
  
  
  
• Zander, N.; Bog, T.; Elhaddad, M.; Frischmann, F.; Kollmannsberger, S.; Rank, E. The multi-level hp-method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes.
  Computer Methods in Applied Mechanics and Engineering 310, pp. 252-277, 2016
  DOI: 10.1016/j.cma.2016.07.007
  
  
  
  
  
  
  
• Zander, N.; Ruess, M.; Bog, T.; Kollmannsberger, S.; Rank, E. Multi-Level hp-Adaptivity for Cohesive Fracture Modelling
  International Journal for Numerical Methods in Engineering 109 (3), pp. 1723-1755, 2016
  DOI: 10.1002/nme.5340 - Status: Verlagsversion / published
  
  
  
  
  
  
  
• Kudela, László; Zander, Nils; Kollmannsberger, Stefan; Rank, Ernst Smart octrees: accurately integrating discontinuous functions in 3D
  Computer Methods in Applied Mechanics and Engineering, 2016 - Status: Verlagsversion / published
  
  
  
  
  
  
  
• Bog, T.; Zander, N.; Kollmannsberger, S.; Rank, E. Show abstract Normal contact with high order finite elements and a fictitious contact material
  Computers & Mathematics with Applications 70 (7), pp. 1370–1390, 2015
  DOI: 10.1016/j.camwa.2015.04.020 - Status: Verlagsversion / published
  
  
  
  Contact problems in solid mechanics are traditionally solved using the h-version of the finite element method. The constraints are enforced along the surfaces of e.g. elastic bodies under consideration. Standard constraint algorithms include penalty methods, Lagrange multiplier methods and combinations thereof. For complex scenarios, a major part of the solution time is taken up by operations to identify points that come into contact. This paper presents a novel approach to model frictionless contact using high order finite elements. Here, we employ an especially designed material model that is inserted into two- respectively three- dimensional regions surrounding contacting bodies. Contact constraints are thus enforced on the same manifold as the accompanying structural problem. The application of the current material formulation leads to a regularization of the Karush-Kuhn-Tucker conditions. Our formulation can be classified as a barrier-type method. Results are obtained for two- and three-dimensional prob- lems, including a Hertzian contact problem. Comparisons to a commercial FEA package are provided. The proposed formulation works well for non-matching discretizations on adjacent contact interfaces and handles self-contact naturally. Since the non-penetrating conditions are solved in a physically consistent man- ner, there is no need for an explicit contact search.
  
  
  
• Kudela, L.; Zander, N.; Bog, T.; Kollmannsberger, S.; Rank, E. Show abstract Efficient and accurate numerical quadrature for immersed boundary methods
  Advanced Modeling and Simulation in Engineering Sciences 2 (1), Springer International Publishing, 2015
  DOI: 10.1186/s40323-015-0031-y - Status: Verlagsversion / published
  
  
  
  One question in the context of immersed boundary or fictitious domain methods is how to compute discontinuous integrands in cut elements accurately. A frequently used method is to apply a composed Gaussian quadrature based on a spacetree subdivision. Although this approach works robustly on any geometry, the resulting integration mesh yields a low order representation of the boundary. If high order shape functions are employed to approximate the solution, this lack of geometric approximation power prevents exponential convergence in the asymptotic range. In this paper we present an algorithmic subdivision approach that aims to be as robust as the spacetree decomposition even for close-to-degenerate cases—but remains geometrically accurate at the same time. Based on 2D numerical examples, we will show that optimal convergence rates can be obtained with a nearly optimal number of integration points.
  
  
  
• Elhaddad, M.; Zander, N.; Kollmannsberger, S.; Shadavakhsh, A.; Nübel, V.; Rank, E. Show abstract Finite Cell Method: High order structural dynamics for complex geometries
  International Journal for Structural Stability and Dynamics 15 (7), pp. 1-25, 2015
  DOI: 10.1142/S0219455415400180 - Status: Verlagsversion / published
  
  
  
  In this contribution, the finite cell method (FCM) is applied to solve transient problems of linear elastodynamics. The mathematical formulation of FCM for linear elastodynamics is presented, following from the weak formulation of the initial/boundary-value problem. Semi-discrete time integration schemes are briefly discussed, and the choice of implicit time integration is justified. A one-dimensional benchmark problem is solved using FCM, illustrating the method's ability to solve problems of linear elastodynamics obtaining high rates of convergence. Furthermore, a numerical example of transient analysis from an industrial application is solved using FCM. The numerical results are compared to the results obtained using state-of-the-art commercial software, employing linear finite elements, in conjunction with explicit time integration. The results illustrate the potential of FCM as a powerful tool for transient analysis in elastodynamics, offering a high degree of accuracy at a moderate computational effort.
  
  
  
• Zander, Nils; Bog, Tino; Kollmannsberger, Stefan; Schillinger, Dominik; Rank, Ernst Show abstract Multi-Level hp-Adaptivity: High-Order Mesh Adaptivity without the Difficulties of Constraining Hanging Nodes
  Computational Mechanics 55 (3), pp. 499ff., 2015
  DOI: 10.1007/s00466-014-1118-x - Status: Verlagsversion / published
  
  
  
  The implementation of hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most hp-code frameworks restrict themselves to 1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the hp-d method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence for problems with non-smooth solutions, is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic hp-adaptivity.
  
  
  
• Zander, N.; Bog, T; Elhaddad, M.; Espinoza, R.; Hu, H.; Joly, A.F.; Wu, C.; Zerbe, P.; Düster, A.; Kollmannsberger, S.; Parvizian, J.; Ruess, M.; Schillinger, D.; Rank, E. FCMLab: A Finite Cell Research Toolbox for MATLAB
  Advances in Engineering Software 74, pp. 49-63, 2014
  DOI: 10.1016/j.advengsoft.2014.04.004 - Status: Verlagsversion / published
  
  
  
  
  
  
  
• Zander, N.; Kollmannsberger, S.; Ruess, M.; Yosibash, Z.; Rank, E. Show abstract The Finite Cell Method for linear thermoelasticity
  Computers & Mathematics with Applications 64 (11), pp. 3527 - 3541, 2012
  DOI: 10.1016/j.camwa.2012.09.002
  
  
  
  The recently introduced Finite Cell Method (FCM) combines the fictitious domain idea with the benefits of high-order Finite Elements. While previous publications concentrated on single-field applications, this paper demonstrates that the advantages of the method carry over to the multi-physical context of linear thermoelasticity. The ability of the method to converge with exponential rates is illustrated in detail with a benchmark problem. A second example shows that the Finite Cell Method correctly captures the thermoelastic state of a complex problem from engineering practice. Both examples additionally verify that, also for two-field problems, Dirichlet boundary conditions can be weakly imposed on non-conforming meshes by the proposed extension of Nitsche's Method. The recently introduced Finite Cell Method (FCM) combines the fictitious domain idea with the benefits of high-order Finite Elements. While previous publications concentrated on single-field applications, this paper demonstrates that the advantages of the method carry over to the multi-physical context of linear thermoelasticity. The ability of the method to converge with exponential rates is illustrated in detail with a benchmark problem. A second example shows that the Finite Cell Method correctly captures the thermoelastic state of a complex problem from engineering practice. Both examples additionally verify that, also for two-field problems, Dirichlet boundary conditions can be weakly imposed on non-conforming meshes by the proposed extension of Nitsche's Method.
  
  
  
• Schillinger, Dominik; Ruess, Martin; Zander, Nils; Bazilevs, Yuri; Düster, Alexander; Rank, Ernst Show abstract Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method
  Computational Mechanics 50 (4), pp. 445-478, Springer-Verlag, 2012
  DOI: 10.1007/s00466-012-0684-z
  
  
  
  The Finite Cell Method (FCM) is an embedded domain method, which combines the fictitious domain approach with high-order finite elements, adaptive integration, and weak imposition of unfitted Dirichlet boundary conditions. For smooth problems, FCM has been shown to achieve exponential rates of convergence in energy norm, while its structured cell grid guarantees simple mesh generation irrespective of the geometric complexity involved. The present contribution first unhinges the FCM concept from a special high-order basis. Several benchmarks of linear elasticity and a complex proximal femur bone with inhomogeneous material demonstrate that for small deformation analysis, FCM works equally well with basis functions of the p-version of the finite element method or high-order B-splines. Turning to large deformation analysis, it is then illustrated that a straightforward geometrically nonlinear FCM formulation leads to the loss of uniqueness of the deformation map in the fictitious domain. Therefore, a modified FCM formulation is introduced, based on repeated deformation resetting, which assumes for the fictitious domain the deformation-free reference configuration after each Newton iteration. Numerical experiments show that this intervention allows for stable nonlinear FCM analysis, preserving the full range of advantages of linear elastic FCM, in particular exponential rates of convergence. Finally, the weak imposition of unfitted Dirichlet boundary conditions via the penalty method, the robustness of FCM under severe mesh distortion, and the large deformation analysis of a complex voxel-based metal foam are addressed.
  
  
  

Conference Proceedings

• Joulaian, Meysam; Zander, Nils; Bog, Tino; Kollmannsberger, Stefan; Rank, Ernst; Düster, Alexander Show abstract A high-order enrichment strategy for the finite cell method
  PAMM 15 (1), pp. 207-208, 2015
  DOI: 10.1002/pamm.201510094
  
  
  
  Abstract Thanks to the application of the immersed boundary approach in the finite cell method, the mesh can be defined independently from the geometry. Although this leads to a significant simplification of the mesh generation, it might cause difficulties in the solution. One of the possible difficulties will occur if the exact solution of the underlying problem exhibits a kink inside an element, for instance at material interfaces. In such a case, the solution turns out less smooth – and the convergence rate is deteriorated if no further measures are taken into account. In this paper, we explore a remedy by considering the partition of unity method. The proposed approach allows to define enrichment functions with the help of a high-order implicit representation of the material interface. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
  
  
  
• Ruess, Martin; Bazilevs, Yuri; Schillinger, Dominik; Zander, Nils; Rank, Ernst Show abstract Weakly enforced boundary conditions for the NURBS-based Finite Cell Method
  In: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), 2012
  
  
  
  In this paper, we present a variationally consistent formulation for the weak enforcement of essential boundary conditions as an extension to the finite cell method, a fictitious domain method of higher order. The absence of boundary fitted elements in fictitious domain or immersed boundary methods significantly restricts a strong enforcement of essential boundary conditions to models where the boundary of the solution domain coincides with the embedding analysis domain. Penalty methods and Lagrange multiplier methods are adequate means to overcome this limitation but often suffer from various drawbacks with severe consequences for a stable and accurate solution of the governing system of equations. In this contribution, we follow the idea of NITSCHE who developed a stable scheme for the solution of the Laplace problem taking weak boundary conditions into account. An extension to problems from linear elasticity shows an appropriate behavior with regard to numerical stability, accuracy and an adequate convergence behavior. NURBS are chosen as a high-order approximation basis to benefit from their smoothness and flexibility in the process of uniform model refinement.
  
  
  
• Rank, E.; Kollmannsberger, S.; Schillinger, D.; Sorger, C.; Zander, N. Show abstract Geometric models, numerical analysis and computational steering in structural engineering
  In: International Conference on Computing in Civil and Building Engineering, 2012
  
  
  
  This paper intends to encourage a discussion on the geometrical and topological description of 3D computational models, and on the embedding of these models into the engineering design process. In the first part, we will report on the development of a mesh generator which tries to overcome some of the present problems of FEM meshers for thin-walled structures and is capable of generating hexahedral meshes for thin, curved geometries as they often appear in civil and mechanical engineering. The second part of the presentation will focus on the recently proposed Finite Cell Method (FCM), a fictitious domain approach, where even very complex geometries of the physical domain can be taken into account without any mesh generation. This new numerical method can be embedded in a computational steering environment for interactive computation, yielding an analysis and design environment, which is far more flexible and intuitive than classical finite elements in civil engineering.
  
  
  

Conference Talks

• Kopp, P.; Korshunova, N.; Seidl, R.; Zander, N.; Kollmannsberger, S.; Rank, E. Finite Cell discretizations for Full Waveform Inversion
  In: European Conference on Computational Mechanics, 2018
  
  
  
  
  
  
  
• Kollmannsberger, S.; Özcan, A.; D'Angella, D.; Carraturo, M.; Kopp, P.; Zander, N.; Reali, A.; Auricchio, F.; Rank, E. Show abstract Computational Modelling of Metal Additive Manufacturing
  In: European Conference on Computational Mechanics, Glasgow, UK, 2018
  
  
  
  The numerical simulation of metal additive manufacturing bears numerous computational challenges. It is a thermo-mechanically coupled process in which material coefficients depend nonlinearly on the state of the material and the temperature. The energy input is highly localized which leads to strong temperature gradients and rapid changes of state in the material on growing computational domains. The large span of the involved spatial and the temporal scales call for highly efficient computational techniques. It is well known that hp-finite elements yield very accurate results for problems with strong gradients or even singular solutions. hp-finite elements are, therefore, an ideal candidate for the simulation of metal additive manufacturing. In this contribution, we present a computational framework which was specifically designed to resolve moving singularities and sharp fronts. Its core employs the multi-level hp-method for the resolution of strong gradients in the solution field. This is complemented by a spatially hierarchic management of material coefficients in the spirit of the finite cell method. We will discuss the computation of the metal additive manufacturing process and evaluate accuracy and efficiency of the presented computational approach by comparison to benchmark solutions.
  
  
  
• Kollmannsberger, S.; Özcan, A.; Carraturo, M.; D'Angella, D.; Zander, N.; Auricchio, A.; Reali, A.; Rank, E. Verification of a multi-level model for the simulation of AM processes
  In: 89. Annual Meeting of the International Assosication of Applied Mathematics and Mechanics, Munich, Germany, 2018
  
  
  
  
  
  
  
• Kopp, P.; Zander, N; Kollmannsberger, S.; Rank, E; Calo V. M. Multilevel-hp adaptive Finite Cell Method for the Navier-Stokes equations using a residual-based Variational Multiscale Method
  In: International Conference on Finite Elements in Flow Problems, Rome, Italy, 2017
  
  
  
  
  
  
  
• Jomo, J.; Zander, N.; Elhaddad, M.; Özcan, A.; Kollmannsberger, S.; Mundani, R-P.; Rank, E. A parallel approach for the multi-level hp-daptive finite cell method
  In: Siam Conference on Computational Science and Engineering 2017, SIAM, 2017
  
  
  
  
  
  
  
• Bog, T.; Zander, N.; Kollmannsberger, S.; Rank, E. Weak imposition of contact constraints on automatically recovered high order embedded interfaces
  In: European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece, 2016
  
  
  
  
  
  
  
• Kollmannsberger, S.; Özcan, A.; Carraturo, M.; Zander, N.; Rank, E. A high-order accurate computational framework for the simulation of SLM processes
  In: European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece, 2016
  
  
  
  
  
  
  
• Kudela, L.; Zander, N.; Kollmannsberger, S.; Rank, E. Smart Octrees: numerical quadrature for immersed boundary methods in three dimensions
  In: European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece, 2016
  
  
  
  
  
  
  
• Zander, N.; D'Angella, D.; Bog, T.; Kollmannsberger, S.; Ruess, M.; Rank, E. Multi-level hp-FEM: dynamic discretizations with arbitrary hanging nodes for solid mechanics
  In: European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece, 2016
  
  
  
  
  
  
  
• Kopp, P.; Zander, N.; Kollmannsberger, S.; Rank, E. Multi-level hp-adaptivity and the finite cell method for fluid problems
  In: European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece, 2016
  
  
  
  
  
  
  
• Jomo, J.; Zander, N.; Elhaddad, M.; Özcan, A.; Kollmannsberger, S.; Mundani, R-P.; Rank, E. Parallelization of the multi-level hp-adaptive finite cell method
  In: 5th European Seminar on Computing, Pilsen, Czech Republic, 2016
  
  
  
  
  
  
  
• Elhaddad, M.; Zander, N.; Jomo, J.; Kollmannsberger, S.; Kirschke, J. S.; Ruess, M.; Rank, E. Show abstract Adaptive discretizations for bone-implant systems using the finite cell method
  In: Engineering Mechanics Institute Conference 2016, Nashville, TN, USA, 2016
  
  
  
  In recent years, the finite element method has been increasingly used to study the biomechanics of bone-implant systems. Patient-specific models need to consider the individual geometry for every case. Typically, geometric models of the bone are reconstructed from medical images --- most commonly CT scans . Following segmentation, a finite element mesh has to be generated. This can be a simple voxel-based mesh, where a predefined number of voxels constitute one finite element in a grid that matches the voxel model. Alternatively, geometry-conforming meshing procedures can be used to generate an unstructured mesh. To this end, the surface points are extracted from the voxel model and converted to a solid model, which is then meshed into an unstructured grid. Similarly, the geometry of the implant is usually reconstructed from medical images, or provided by Computer-Aided-Design (CAD) models, and then incorporated in the finite element mesh. The adequate treatment of the bone-implant interface demands the spatial refinement of the finite element mesh at the interface for an accurate solution. For voxel-based finite elements, a very fine resolution is already necessary to resolve the implant's geometry. This leads to an extremely high number of degrees of freedom, as the whole grid has to be refined. On the other hand, geometry-conforming finite elements need more pre-processing (converting the bone's voxel model to a solid model), and complex meshing procedures to resolve geometries of the bone and the implant with graded refinement of the mesh towards the interface. In this contribution, we will use the finite cell method (FCM) in conjunction with a hierarchical refinement scheme to model a vertebral fixation system. The FCM discretization completely circumvents the mesh generation procedure, as the geometry is resolved on the integration level. Here, the flexibility of FCM in dealing with different types of geometric representations is demonstrated: the bone's geometry is represented by the voxel model from a CT-scan, whereas the implant's geometry is directly described by a CAD solid model. Both representations are directly used in the FCM framework, thereby substantially reducing the pre-processing effort. Furthermore, the hierarchical refinement scheme yields a higher accuracy at the bone-implant interface by adaptively refining the FCM grid.
  
  
  
• Elhaddad, M.; Zander, N.; Kollmannsberger, S.; Bauer, J.; Ruess, M.; Rank, E. Show abstract Loading simulation of spinal vertebrae using the finite cell method
  In: VI International Conference on Computational Bioengineering, Barcelona, Spain, 2015
  
  
  
  Osteoporosis compromises bone strength, increasing the risk of vertebral fractures with severe health consequences. The development of an accurate and reliable patient-specific vertebral model would be of major clinical relevance, for both the prognosis of fractures and the investigation of implant systems. In the case of spinal fusion, the peri-implant bone strains are of major interest. High forces are transmitted at the bone-implant interface and screw loosening is the major cause of surgical failure. In recent years, the finite element method has been increasingly applied to predict the biomechanical response of vertebrae. In general, quantitative CT-data is used to retrieve the geometric model and the inhomogeneous material properties. A finite element mesh is then generated for the numerical simulation. For the finite element analysis of a bone-implant system, the adequate treatment of the bone-implant interface demands the adaptive refinement of the finite element mesh for an accurate solution. In this contribution, the finite cell method (FCM), a high order embedded domain approach, is applied to simulate the biomechanical behavior of human vertebrae. The recently developed multi- level hierarchical refinement is used to enrich the FCM, allowing to resolve the highly inhomogeneous material properties of vertebrae. For the simulation of bone-implant systems, the hierarchic refinement is used to resolve the bone-implant interface appropriately. The feasibility of using FCM for predicting the biomechanical response of human vertebrae is demonstrated with numerical examples on both intact and fused vertebral segments. Furthermore, the inherent hierarchical nature of the FCM strongly supports verification and validation.
  
  
  
• Rank, E.; Bog, T.; Elhaddad, M.; Kollmannsberger, S.; L., Kudela; Zander, N. Show abstract The Finite Cell Method: Some Principles and Recent Progress
  In: 13th U.S. National Congress on Computational Mechanics, San Diego, California, USA, 2015
  
  
  
  The finite cell method (FCM) is a combination of a fictitious domain approach with finite elements of high order. The physical domain is embedded into an easy to mesh bigger computational domain. Finite cell meshes are then generated ignoring boundaries of the physical domain and result in rectangular ‘cells’ as support for the higher order shape functions. The geometry of the problem is only considered during the integration of the cell matrices. To this end an indicator function is introduced being equal to 1 inside the physical domain. Outside it is set (in order to avoid conditioning problems) to a very small value. The contribution of the fictitious domain is thus penalized, shifting the effort of meshing towards the numerical integration of the cell matrices. Combining these ingredients, an exponential rate of convergence can be observed for smooth problems, when performing a p-extension. Since the quality and efficiency of the finite cell approximation strongly depends on the numerical integration scheme, this presentation will first discuss a new algorithmic subdivision approach, extending the conventional octree-based integration with the ability of resolving ‘kinks’ or corners of the physical domain. In combination with the blending function method, this algorithm yields a nearly exact decomposition of the cut cells. Our approach is able to resolve close-to-degenerate cases, but remains algorithmically simple at the same time. Several further recent developments of the FCM will be discussed. A new two- and three-dimensional hierarchical refinement strategy yields exponential rate of convergence in energy norm even for singular problems, and its simple algorithmic structure allows an easy extension to transient problems with local refinement and de-refinement. Finally, a re-interpretation of the fictitious domain in the sense of a ‘third material’ opens the way for a modified contact formulation without the necessity of contact search.
  
  
  
• Bog, T.; Zander, N.; Kollmannsberger, S.; Rank, E. Small and large deformation contact simulation using the finite cell method
  In: European Solid Mechanics Conference 2015 (ESMC2015), Madrid, Spain, 2015
  
  
  
  
  
  
  
• Rank, E.; Elhaddad, M.; Zander, N.; Kollmannsberger, S.; Bauer, J.; Ruess, M. Show abstract Coupling scales and adaptive refinement for simulation of bone-fixation interfaces using the Finite Cell Method
  In: VI International Conference on Coupled Problems in Science and Engineering, Venice, Italy, 2015
  
  
  
  In recent years many studies on the numerical simulation of bone-implant systems have been presented. In general, a geometric model of a bone is derived from qCT-data, including its highly inhomogeneous material properties. Bone and implant are then meshed and simulated by finite elements. In the case of a fixation the zone of the bone close to the screw is of major interest. High forces are transmitted here and screw loosening is the major cause of surgical failure. As the scale of interest for this interface region may be several orders of magnitude smaller than that of the bone, an adaptive refinement of the finite element mesh is mandatory for an accurate solution. Whereas this is already quite demanding in cases of a single simulation, it becomes prohibitively expensive if a sequence of many simulations in order to optimize type, position and pre-stress of an instrumentation shall be performed.In this paper we will extend the recently developed Finite Cell Method, a high order embedded domain approach by a hierarchical refinement strategy in the regions of interest. Whereas the concept of the FCM completely relieves from the need to generate a finite element mesh, the hierarchical refinement yields an accuracy and a scale resolution which could not obtained by classical finite elements methods. We will discuss the basic properties of this adapted multi-scale method and demonstrate its feasibility for simulating bone-mechanics on the example of the fixation of a spinal segment.
  
  
  
• Zander, Nils; Tino, Bog; Kollmannsberger, Stefan; Ruess, Martin; Schillinger, Dominik; Rank, Ernst Multi-level hp-adaptivity for transient discretizations of complex domains
  In: Engineering Mechanics Institute Conference, Stanford, USA, 2015
  
  
  
  
  
  
  
• Nübel, V.; Shadavakhsh, A.; Elhaddad, M.; Zander, N.; Kollmannsberger, S. Dynamic analysis of high loaded components, discretized by fictitious domain methods
  In: 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014
  
  
  
  
  
  
  
• Zander, N.; Bog, T..; Kollmannsberger, S.; Schillinger, D.; Rank, E. High-Order hp-FEM: High Order Mesh Adaptivity without the difficulties of Hanging Nodes
  In: 11th. World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014
  
  
  
  
  
  
  
• Bog, Tino; Zander, Nils; Kollmannsberger, Stefan; Rank, Ernst A contact formulation based on high order fictitious domain methods
  In: 11th. World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014
  
  
  
  
  
  
  
• Bog, T.; Zander, N.; Kollmannsberger, S.; Rank, E. The Finite Cell Method for Contact Problems in Solid Mechanics
  In: 3rd International Conference on Computational Contact Mechanics (ICCCM 2013), Lecce, Italy, 2013
  
  
  
  
  
  
  
• Bog, T.; Zander, N.; Kollmannsberger, S.; Rank, E. The Finite Cell Method for Contact Problems in Solid Mechanics
  In: 6th EUROPEAN CONGRESS ON COMPUTATIONAL METHODS IN APPLIED SCIENCES AND ENGINEERING (ECCOMAS), 2012
  
  
  
  
  
  
  
• Zander, N.; Erbts, P.; Kollmannsberger, S.; Düster, A.; Rank, E. Show abstract The Finite Cell Method for Transient, Non-linear Heat Conduction
  In: European Seminar on Computing, Pilsen, Czech Republic, 2012
  
  
  
  Since the early years of computational engineering, the classical Finite Element Method (FEM) has become the state-of-the-art approach to solve initial boundary value problems numerically. Although major enhancements allowed for highly sophisticated simulations, the idea to geometrically resolve the physical domain on the discretization level remained unchanged. In cases of complex geometries, this intrinsic need for a conform mesh is still a burden in today’s engineering practice. The recently introduced Finite Cell Method (FCM) overcomes this problem by combining the benefits of high-order Finite Elements (p-FEM) with the idea of fictitious domains. The approach embeds the possibly complex physical domain phys in a fictitious domain ?fict and solves the problem on their simply shaped union. The localization factor ? allows to recover the original geometry on the integration level. Recent research results confirm the excellent applicability of this new method in the fields of non-linear continuum mechanics, topology optimization, thin-walled structures, bone mechanics and advection-diffusion problems. Also the method’s potential for multi-physical problems has been demonstrated in the context of linear thermoelasticity. In the present work, the Finite Cell Method is employed to solve the transient, non-linear heat equation on non-conforming meshes. In particular, the weak enforcement of Dirichlet boundary conditions is addressed. The results of a conventional p-FEM simulation serve as reference and the convergence characteristics of both approaches are compared.
  
  
  
• Kollmannsberger, S.; Zander, N.; Monavari, M.; Ruess, M.; Düster, A.; Rank, E. The Finite Cell Method for imaged based modeling of heterogeneous materials
  In: ECCOMAS, 2012
  
  
  
  
  
  
  

Poster

• Davide D'Angella; Nils Zander; Stefan Kollmannsberger; Felix Frischmann; Andreas Schröder; Ernst Rank; Alessandro Reali Explicit Error Estimation and Multi-Level hp-Adaptivity
  In: Seventh International Workshop on High-Order Finite Element and Isogeometric Methods, Jerusalem, Israel, 2016
  
  
  
  
  
  
  
• Bog, Tino; Zander, Nils; Kollmannsberger, Stefan; Rank, Ernst A formulation for frictionless contact using a material model and high order finite elements
  In: Sixth International Workshop on High-Order Finite Element and Isogeometric Methods, Frauenchiemsee Island, Germany, 2014
  
  
  
  
  
  
  
• Zander, Nils; Bog, Tino; Kollmannsberger, Stefan; Schillinger, Dominik; Rank, Ernst Multi-Level hp-Adaptivity: High-Order Mesh Adaptivity without the Difficulties of Constraining Hanging Nodes
  In: Sixth International Workshop on High-Order Finite Element and Isogeometric Methods, Frauenchiemsee Island, Germany, 2014
  
  
  
  
  
  
  

Theses

• Zander, Nils Show abstract The Finite Cell Method for Linear Thermoelasticity
  Master's thesis, Technische Universität München, 2011
  
  
  
  This thesis presents an approach to approximate the phenomena of linear thermoelasticity numerically in the framework of the Finite Cell Method (FCM). For this purpose, the governing equations of the thermal, elastic and linear thermoelastic system are derived in their strong and weak formulation. In particular, it is outlined how Dirichlet boundary conditions of the three problems can be imposed in the weak sense. In the second part of this work, the constrained weak field equations are discretized in space utilizing the ideas of the Finite Cell Method. The performance of this approach is demonstrated through different examples.
  
  
  

Teaching

Supervised Theses

• Kopp, P Multi-level hp-FEM and the Finite Cell Method for the Navier-Stokes equations using a Variational Multiscale Formulation
  Master's thesis, Technische Universität München, 2017
  
  
  
  
  
  
  
• Korshunova, Nina Partitioned hp-d approach for multiscale transient heat problems
  Master's thesis, Technische Universität München, 2016
  
  
  
  
  
  
  
• Coradello, Luca Implementation of a high-order Kirchhoff-Love shell: a comparison of IGA and p-FEM
  Master's thesis, Technische Universität München, 2016
  
  
  
  
  
  
  
• D'Angella, D. Show abstract A Posteriori Error Estimator for the Multi-Level hp-Finite Element Method
  Master's thesis, Technische Universität München, 2015
  
  
  
  This works aims to investigate the applicability of residual-based error estimators to two variants of the Finite Element Method: the multi-level Finite Element Method and the Finite Cell's Method (FCM). The study is performed both theoretically and numerically, showing excellent results for the Multi-Level Finite Element Method in the context of Poisson's and linear elastic problems. Furthermore, the analysis carried out in the context of the FCM furnishes a starting point for further developments and investigations on how to estimate the error in case of a non-conforming discretization.
  
  
  
• Buckel, Sebastian Dynamic Substructuring using the Finite Cell Method
  Bachelor's thesis, Technische Universität München, 2014
  
  
  
  
  
  
  
• Dumonet, Dimitri Towards Efficient and Accurate 3D Cut Cell Integration in the Context of the Finite Cell Method
  Bachelor's thesis, Technische Universität München, 2014
  
  
  
  
  
  
  
• Kopp, Philipp Implementation of the Characteristic Based Split algorithm for the Shallow Water Equations in a high order finite element frame-work
  Bachelor's thesis, Technische Universität München, 2014
  
  
  
  
  
  
  
• Elhaddad, M. Show abstract Transient Analysis with the Finite Cell Method
  Master's thesis, Technische Universität München, 2014
  
  
  
  In the thesis at hand, the finite cell method (FCM) is applied to solve transient problems of linear elastodynamics. The mathematical formulation of FCM for linear elastodynamics is presented, following from the weak formulation of the initial/boundary-value problem. Semi-discrete time integration schemes are briefly discussed, and the choice of implicit time integration is justified. A simple benchmark problem is solved using FCM, illustrating the method's abiltiy to solve problems of linear elastodynamics with high convergence rates. Finally, a numerical example of transient analysis from an industrial application is solved using FCM. The numerical results are compared to a solution obtained using state-of-the-art commercial software, employing linear finite elements, in conjunction with explicit time integration. The results illustrate the potential of FCM as a powerful tool for transient analysis in elastodynamics.
  
  
  
• Miksch, Matthias Show abstract Acoustic source localization based on time reversed simulations with high order Finite-Element Methods
  Bachelor's thesis, Technische Universität München, 2014
  
  
  
  This bachelor’s thesis depicts the source localization of acoustic source in solid bodies. For this purpose two fundamental equations are introduced, the equation of motion and the wave equation. Both equations were discretized in space with the Finite Element Method and in time with the Newmark’s scheme. Beside the simulation of a wave that spreads out in space, it is possible to simulate the wave in time-reverse such that the wave propagates back to its origin. The equations were solved numerically within MATLAB. Therefore an existing finite element implementation, which was provided by the chair of Computation in Engineering of the Technische Universität München, was extended.
  
  
  
• Kudela, László Show abstract Highly Accurate Subcell Integration in the Context of The Finite Cell Method
  Master's thesis, Technische Universität München, 2013
  
  
  
  The Finite Cell Method is an immersed boundary method combined with higher order finite elements. In FCM, the material parameters of the fictitious domain are scaled down, thus a discontinuity is introduced in the cut finite cells. This introduces an error in the integrals of the stiffness matrix. Typically, composed Gaussian quadrature for evaluating integrals in the cut cells is used, based on a spacetree decomposition. The drawback of the spacetree decomposition is that it introduces a high number of quadrature points and yields only a 0 th order aproximation of the geometry. The thesis proposes a different apporach to deal with integration: the cut cells are decomposed into quadrilaterals and triangles with curved sides and the quadrature points are distributed on their parameter spaces. To this end, an algorithm is proposed that performs this decomposition in two dimensions. The algorithm is able to work on complex geometries. The results show that the blended subcell integration is able to produce results with higher precision yet still less integration points than the spacetree approach. Thus, the overall computational effort of FCM analyses is considerably reduced in comparison to the spacetree integration. This makes the proposed approach a good fit into the design-through-analysis pipeline.
  
  
  
• Ruchti, Martin Show abstract Model Simplification via Super Elements using the Finite Cell Method
  Bachelor's thesis, Technische Universität München, 2013
  
  
  
  In this bachelor’s thesis, the theory and the application of the super element approach in the Finite Cell Method is given. Furthermore, an implementation in the FCM framework AdhoC++, which was developped by the chair of Computation in Engineering of the Technische Universität München is introduced. For simplicity, linear elasticity, static analysis and two-dimensionality are assumed. For generating super elements, weak constraints are used. The program developped as part of this thesis is analyzed by several tests, comparing results from a classical mesh patch test with results from a super element, obtained by patching meshes involving a reduced mesh.
  
  
  
• von Danwitz, Max Automated application of Dirichlet boundary conditions in voxel based analyses using the Finite Cell Method
  Bachelor's thesis, Technische Universität München, 2013
  
  
  
  
  
  
  

Courses

• Bau- und Umweltinformatik 2 (Summer Term 14)
• Algorithmik partieller Differentialgleichungen (Winter Term 13/14)
• Introduction to Programming in C++ (Winter Term 11/12, 12/13, 13/14)

Supervised Software Labs

• FCM Toolbox for MATLAB, Software Lab, 2012