Computational Mathematics

Department of Mathematics at Humboldt Universität zu Berlin

Reviewed Journal Articles

D. Knees and A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35(15):1859-1184, 2012
[ pdf | url | ]
A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B 3/2 2,∞ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.

@article{Knees2011M2AS,
  author = {Knees, Dorothee and Schr{\"o}der, Andreas},
  title = {Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints},
  journal = {Mathematical Methods in the Applied Sciences},
  volume= {35(15)},
  pages= {1859-1184},
  year = 2012,
  pdf = {http://www.wias-berlin.de/preprint/1673/wias_preprints_1673.pdf},
  url = {http://onlinelibrary.wiley.com/doi/10.1002/mma.2598/pdf},
  abstract = {A global higher differentiability result in Besov spaces is proved for the displacement  fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts  (3D) the displacement fields are B 3/2  2,∞ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied  to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems  with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities,  convergence rates for FE-discretizations of contact problems are derived relying on  the proven regularity properties. Several numerical examples illustrate the theoretical results.}
}
H. Blum, A. Rademacher and A. Schröder, Goal oriented error control for frictional contact problems in metal forming, Key Engineering Materials, 504-506:987-992, 2012
[ url | ]
In this note, techniques for goal oriented error control of finite element discretizations are proposed for frictional contact problems. The finite element discretization is based on a mixed method, where Lagrange multipliers are introduced to capture the geometrical and frictional contact conditions. A posteriori error estimates for user-defined, probably non-linear quantities of interest are derived using the dual weighted residual method (DWR). Numerical results substantiate the applicability of the presented techniques to the simulation of metal forming processes.

@article{BRS11KEM,
  author = {Blum, Heribert and Rademacher, Andreas and Schr{\"o}der, Andreas},
  title = {Goal oriented error control for frictional contact problems in metal forming},
  abstract = {In this note, techniques for goal oriented error control of finite element discretizations are proposed for frictional contact problems. The finite element discretization is based on a mixed method, where Lagrange multipliers are introduced to capture the geometrical and frictional contact conditions. A posteriori error estimates for user-defined, probably non-linear quantities of interest are derived using the dual weighted residual method (DWR). Numerical results substantiate the applicability of the presented techniques to the simulation of metal forming processes.}
  journal = {Key Engineering Materials},
  volume = {504-506},
  pages = {987-992},
  year = 2012,
  url = {http://www.scientific.net/KEM.504-506.987}
}
A. Schröder, A posteriori error estimates of higher-order finite elements for frictional contact problems, Computer Methods in Applied Mechanics and Engineering, 249-252:151-157, 2012
[ doi | pdf | ]
In this paper, a posteriori estimates are derived for higher-order finite element methods and frictional contact problems. The discretization is based on a mixed approach where the geometrical and frictional constraints are captured by Lagrange multipliers. The use of higher-order polynomials leads to a certain non-conformity in the discretization which requires special attention in the error analysis. As a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms. The residual is estimated by well-known a posteriori error estimates for variational equations. The remainder terms represent typical sources resulting from the non-conforming mixed discretization. Numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements.

@article{Schroeder12CMAME,
  author = {Schr{\"o}der, Andreas},
  title = {A~posteriori error estimates of higher-order finite elements for frictional contact problems},
  journal = {Computer Methods in Applied Mechanics and Engineering},
  volume = {249-252},
  pages = {151-157}
  year = 2012,
  abstract = {In this paper, a posteriori estimates are derived for higher-order finite element methods and frictional contact problems. The discretization is based on a mixed approach where the geometrical and frictional constraints are captured by Lagrange multipliers. The use of higher-order polynomials leads to a certain non-conformity in the discretization which requires special attention in the error analysis. As a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms. The residual is estimated by well-known a posteriori error estimates for variational equations. The remainder terms represent typical sources resulting from the non-conforming mixed discretization. Numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements.},
  keywords = {contact problems, friction, higher-order finite elements, error estimates},
  doi = {http://dx.doi.org/10.1016/j.cma.2012.02.001},
  pdf = {http://www2.mathematik.hu-berlin.de/publ/pre/2011/P-11-24.pdf}
}
A. Schröder and S. Wiedemann, Error estimates in elastoplasticity using a mixed method, Applied Numerical Mathematics, 61:1031-1045, 2011
[ doi | pdf | ]
In this paper, a mixed formulation and its discretization are introduced for elastoplasticity with linear kinematic hardening. The mixed formulation relies on the introduction of a Lagrange multiplier to resolve the non-differentia­bility of the plastic work function. The main focus is on the derivation of a priori and a posteriori error estimates based on general discretization spaces. The estimates are applied to several low-order finite elements. In particular, a posteriori estimates are expressed in terms of standard residual estimates. Numerical experiments are presented, confirming the applicability of the a posteriori estimates within an adaptive procedure.

@article{Schroeder2011,
  author = {Schr{\"o}der, Andreas and Wiedemann, Sebastian},
  title = {Error estimates in elastoplasticity using a mixed method},
  journal = {Applied Numerical Mathematics},
  year = 2011,
  volume = {61},
  pages = {1031--1045},
  abstract = {In this paper, a mixed formulation and its discretization are introduced for elastoplasticity with linear kinematic hardening. The mixed formulation relies on the introduction of a Lagrange multiplier to resolve the non-differentia\-bility of the plastic work function. The main focus is on the derivation of a~priori and a~posteriori error estimates based on general discretization spaces. The estimates are applied to several low-order finite elements. In particular, a~posteriori estimates are expressed in terms of standard residual estimates. Numerical experiments are presented, confirming the applicability of the a~posteriori estimates within an adaptive procedure.},
  doi = {10.1016/j.apnum.2011.06.001},
  pdf = {http://www2.mathematik.hu-berlin.de/publ/pre/2011/P-11-01.pdf}
}
A. Schröder and A. Rademacher, Goal-oriented error control in adaptive mixed FEM for Signorini's Problem., Comp. Meth. Appl. Mech. Engrg., 200(1-4):345-355, 2011
[ doi | url | ]
This paper outlines goal-oriented finite element error control for Signorini’s problem. The discretization is based on a mixed formulation proposed by Hlavacek et al. which is extended to higher-order polynomials. A duality argument based on a variational inequality is applied, which allows for the estimates in h- as well as hp-adaptivity. Numerical results confirm the applicability of the theoretical findings.

@article{Schroeder09CMAME,
  author = {Schr{\"o}der, Andreas and Rademacher, Andreas},
  title = {{Goal-oriented error control in adaptive mixed FEM for Signorini's Problem.}},
  journal = {Comp. Meth. Appl. Mech. Engrg.},
  volume = {200},
  number = {1-4},
  pages = {345-355},
  year = 2011,
  abstract = {This paper outlines goal-oriented finite element error control for Signorini’s problem. The discretization is based on a mixed formulation proposed by Hlavacek et al. which is extended to higher-order polynomials. A duality argument based on a variational inequality is applied, which allows for the estimates in h- as well as hp-adaptivity. Numerical results confirm the applicability of the theoretical findings.},
  url = {http://www.sciencedirect.com/science/article/pii/S0045782510002483}
  doi = {10.1016/j.cma.2010.08.015}
}
A. Schröder, H. Blum, A. Rademacher and H. Kleemann, Mixed FEM of higher order for contact Problems with friction., Journal Numerical Analysis & Modeling, 8(2):302-323, 2011
[ pdf | ]
This paper presents mixed variational formulation and its discretization with finite elements of higher-order for Signorini’s problem with Tresca’s friction. To guarantee the unique existence of the discrete saddle point of the mixed method, a discrete inf-sup condition is proven. Moreover, a solution scheme based on the dual formulation of the mixed method is proposed. Numerical results confirm the theoretical findings.

@article{Schroeder10JNAM,
  author = {Schr{\"o}der, Andreas and Blum, Heribert and Rademacher, Andreas and Kleemann, Heiko},
  title = {{Mixed FEM of higher order for contact Problems with friction.}},
  journal = {Journal Numerical Analysis \& Modeling},
  volume = {8},
  number = {2},
  pages = {302-323},
  abstract = {This paper presents mixed variational formulation and its discretization with finite elements of higher-order for Signorini’s problem with Tresca’s friction. To guarantee the unique existence of the discrete saddle point of the mixed method, a discrete inf-sup condition is proven. Moreover, a solution scheme based on the dual formulation of the mixed method is proposed. Numerical results confirm the theoretical findings.},
  year = 2011,
  pdf = {http://www.math.hu-berlin.de/~compmath/publications/../downloads/jnam_schroeder.pdf}
}
A. Schröder, Mixed finite element methods of higher-order for model contact problems, SINUM J. Numer. Anal. 49(6): 2323-2339, 2011
[ pdf | ]
This paper presents mixed finite element methods of higher-order for a simplified Signorini problem and an idealized frictional problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. To guarantee the unique existence of the solution of the mixed method, a discrete inf-sup condition is proven. Approximation results of the p-method of finite elements and some inverse estimates for higher-order polynomials are applied. Numerical results confirm the theoretical findings.

@article{Schroeder09SIN,
  author = {Schr{\"o}der, Andreas},
  title = {{Mixed finite element methods of higher-order for model contact problems}},
  journal = {SINUM J. Numer. Anal. 49(6): 2323-2339},
  year = 2011,
  abstract = {This paper presents mixed finite element methods of higher-order for a simplified Signorini problem and an idealized frictional problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. To guarantee the unique existence of the solution of the mixed method, a discrete inf-sup condition is proven. Approximation results of the p-method of finite elements and some inverse estimates for higher-order polynomials are applied. Numerical results confirm the theoretical findings.},
  pdf = {http://www.math.hu-berlin.de/~compmath/publications/../downloads/p-09-16.pdf}
}
D. Knees and A. Schröder, Computational aspects of quasi-static crack propagation, Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 6(1):64-99, 2013
[ pdf | url | ]
The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.

@article{Knees2011WIAS,
  author = {Knees, D. and Schr{\"o}der, Andreas},
  title = {Computational aspects of quasi-static crack propagation},
  journal = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)},
  volume= {6(1)},
  pages={64-99},
  year = 2013,
  abstract = {The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.},
  url = {http://wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=2011&number=1611},
  pdf = {http://www.wias-berlin.de/preprint/1611/wias_preprints_1611.pdf}
}
A. Byfut and A. Schröder, hp-Adaptive Extended Finite Element Method., International Journal for Numerical Methods in Engineering, published online, 89(11):1392-1418, 2012
[ doi | url | ]
This paper discusses higher-order extended finite element methods obtained from the combination of the standard extended finite element method (XFEM) with higher-order finite element methods. Here, the focus is on the embedding of the latter into the partition of unity method, which is the basis of the XFEM. A priori error estimates are discussed and numerical verification is given for two benchmark problems. Moreover, methodological aspects are summarized which are necessary for hp-adaptivity in XFEM and allow for exponential convergence rates. In particular, the handling of hanging nodes via constraint approximation and an hp-adaptive strategy are presented.

@article{Byfut10IJNME,
  author = {Byfut, Andreas and Schr{\"o}der, Andreas},
  title = {{$hp$-Adaptive Extended Finite Element Method.}},
  journal = {International Journal for Numerical Methods in Engineering, published online},
  year = 2012,
   volume = {89},
  number = {11},
  pages = {1392-1418},
  abstract = {This paper discusses higher-order extended finite element methods obtained from the combination of the standard extended finite element method (XFEM) with higher-order finite element methods. Here, the focus is on the embedding of the latter into the partition of unity method, which is the basis of the XFEM. A priori error estimates are discussed and numerical verification is given for two benchmark problems. Moreover, methodological aspects are summarized which are necessary for hp-adaptivity in XFEM and allow for exponential convergence rates. In particular, the handling of hanging nodes via constraint approximation and an hp-adaptive strategy are presented.},
  url = {http://onlinelibrary.wiley.com/doi/10.1002/nme.3293/abstract},
  doi = {10.1002/nme.3293}
}
F. Weichert, A. Schröder, C. Landes, L. Walczak, H. Müller and M. Wagner, Finite element simulation of skeletal muscular structures obtained from images of histological serial sections., English, Journal of Biomechanics, 43(8):1483-1487, May) 2010
[ doi | ]
In this study, we present a method for the three-dimensional reconstruction of objects obtained from histological serial sections (exemplified by those of a pennate striated skeletal muscle) and its application to the finite element method. A hyperelastic material model is used for modeling biological soft tissue. The reconstruction process relies on the direct construction of a volumetric mesh using an octree approach which leads to a stable finite element method. Stability can be expressed in the spectral matrix condition number. To visualize stress patterns within the underlying anatomy the simulation results are projected onto images of the histological scenario.

@article{Weichert10JB,
  author = {Weichert, Frank and Schr{\"o}der, Andreas and Landes, C. and Walczak, Lars and M{\"u}ller, H. and Wagner, M.},
  title = {Finite element simulation of skeletal muscular structures obtained from images of histological serial sections.},
  language = {English},
  journal = {Journal of Biomechanics},
  year = 2010,
  volume = {43},
  number = {8},
  pages = {1483-1487},
  month = {May},
  pmid = {20181337},
  issn = {0021-9290},
  keywords = {Mesh generation; Finite element simulation; Histological cuts; Systems biology; Cleft lip and palate},
  abstract = {In this study, we present a method for the three-dimensional reconstruction of objects obtained from histological serial sections (exemplified by those of a pennate striated skeletal muscle) and its application to the finite element method. A hyperelastic material model is used for modeling biological soft tissue. The reconstruction process relies on the direct construction of a volumetric mesh using an octree approach which leads to a stable finite element method. Stability can be expressed in the spectral matrix condition number. To visualize stress patterns within the underlying anatomy the simulation results are projected onto images of the histological scenario.},
  doi = {10.1016/j.jbiomech.2010.01.046}
}
F. Weichert, A. Schröder, C. Landes, A. Shamaa, S. K. Awad, L. Walczak, H. Müller and M. Wagner, Computation of a finite element-conformal tetrahedral mesh approximation for simulated soft tissue deformation using a deformable surface model, Med Biol Eng Comput, 48(6):597-610, Apr) 2010
[ doi | ]
In this article, we present a new method for the generation of surface meshes of biological soft tissue. The method is based on the deformable surface model technique and is extended to histological data sets. It relies on an iterative adjustment towards polygonal segments describing the histological structures of the soft tissue. The generated surface meshes allow for the construction of volumetric meshes through a standard constrained Delaunay approach and, thus, for the application in finite element methods. The geometric properties of volumetric meshes have an immediate influence on the numerical conditioning and, therewith, on the stability of the finite element method and the convergence of iterative solvers. In this article, the influence of the surface meshes on the quality of the volumetric meshes is analysed in terms of the spectral condition number of the stiffness matrices, which are assembled within Newton's method. The non-linear material behavior of biological soft tissue is modeled by the Mooney-Rivlin material law. The subject is motivated by the requirements of virtual surgery.

@article{Weichert10MBEC,
  author = {Weichert, Frank and Schr{{\"o}}der, A. and Landes, C. and Shamaa, A. and Awad, S. K. and Walczak, Lars and M{{\"u}}ller, H. and Wagner, M.},
  title = {{Computation of a finite element-conformal tetrahedral mesh approximation for simulated soft tissue deformation using a deformable surface model}},
  abstract = {In this article, we present a new method for the generation of surface meshes of biological soft tissue. The method is based on the deformable surface model technique and is extended to histological data sets. It relies on an iterative adjustment towards polygonal segments describing the histological structures of the soft tissue. The generated surface meshes allow for the construction of volumetric meshes through a standard constrained Delaunay approach and, thus, for the application in finite element methods. The geometric properties of volumetric meshes have an immediate influence on the numerical conditioning and, therewith, on the stability of the finite element method and the convergence of iterative solvers. In this article, the influence of the surface meshes on the quality of the volumetric meshes is analysed in terms of the spectral condition number of the stiffness matrices, which are assembled within Newton's method. The non-linear material behavior of biological soft tissue is modeled by the Mooney-Rivlin material law. The subject is motivated by the requirements of virtual surgery.},
  journal = {Med Biol Eng Comput},
  volume = {48},
  number = {6},
  pages = {597-610},
  year = 2010,
  month = {Apr},
  pmid = {20411435},
  doi = {10.1007/s11517-010-0607-0}
}
T. Fries, A. Byfut, A. Alizada, A. Cheng and A. Schröder, Hanging nodes and XFEM., Int. J. Numer. Methods Eng., 86(4-5):404-430, 2011
[ doi | url | ]
This paper investigates two approaches for the handling of hanging nodes in the framework of extended finite element methods (XFEM). Allowing for hanging nodes, locally refined meshes may be easily generated to improve the resolution of general, i.e. model-independent, steep gradients in the problem under consideration. Hence, a combination of these meshes with XFEM facilitates an appropriate modeling of jumps and kinks within elements that interact with steep gradients. Examples for such an interaction are, e.g. found in stress fields near crack fronts or in boundary layers near internal interfaces between two fluids. The two approaches for XFEM based on locally refined meshes with hanging nodes basically differ in whether (enriched) degrees of freedom are associated with the hanging nodes. Both approaches are applied to problems in linear elasticity and incompressible flows.

@article{Fries10IJNME,
  author = {Fries, T.P. and Byfut, Andreas and Alizada, A. and Cheng, A. and Schr{\"o}der, Andreas},
  title = {{Hanging nodes and XFEM.}},
  journal = {Int. J. Numer. Methods Eng.},
  year = 2011,
  volume = {86},
  number = {4-5},
  pages = {404-430},
  abstract = {This paper investigates two approaches for the handling of hanging nodes in the framework of extended finite element methods (XFEM). Allowing for hanging nodes, locally refined meshes may be easily generated to improve the resolution of general, i.e. model-independent, steep gradients in the problem under consideration. Hence, a combination of these meshes with XFEM facilitates an appropriate modeling of jumps and kinks within elements that interact with steep gradients. Examples for such an interaction are, e.g. found in stress fields near crack fronts or in boundary layers near internal interfaces between two fluids. The two approaches for XFEM based on locally refined meshes with hanging nodes basically differ in whether (enriched) degrees of freedom are associated with the hanging nodes. Both approaches are applied to problems in linear elasticity and incompressible flows.},
  url ={http://onlinelibrary.wiley.com/doi/10.1002/nme.3024/abstract},
  doi = {10.1002/nme.3024}
}
A. Schröder, Error control in h- and hp-adaptive FEM for Signorini's problem., English, J. Numer. Math., 17(4):299-318, 2009
[ doi | ]
Summary: This paper presents a posteriori finite element error estimates for Signorini's problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The a posteriori error control relies on estimating the discretization error of an auxiliary problem which is given as a variational equation. The estimation consists of error bounds for the discretization error of the auxiliary problem and some further terms which capture the geometrical error and the error in the complementary condition. The derived estimates are applied to h- and hp-adaptive refinement and enrichment strategies. Numerical results confirm the applicability of the theoretical findings. In particular, optimal algebraic and almost exponential convergence rates are obtained.

@article{Schroeder09JNM,
  author = {Schr{\"o}der, Andreas},
  title = {{Error control in $h$- and $hp$-adaptive FEM for Signorini's problem.}},
  language = {English},
  journal = {J. Numer. Math.},
  volume = {17},
  number = {4},
  pages = {299-318},
  year = 2009,
  doi = {10.1515/JNUM.2009.015},
  abstract = {Summary: This paper presents a posteriori finite element error estimates for Signorini's problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The a posteriori error control relies on estimating the discretization error of an auxiliary problem which is given as a variational equation. The estimation consists of error bounds for the discretization error of the auxiliary problem and some further terms which capture the geometrical error and the error in the complementary condition. The derived estimates are applied to $h$- and $hp$-adaptive refinement and enrichment strategies. Numerical results confirm the applicability of the theoretical findings. In particular, optimal algebraic and almost exponential convergence rates are obtained.},
  keywords = {$hp$-FEM; contact problems; error control},
  classmath = {*74-99 (Mechanics of deformable solids)}
}
H. Blum, A. Rademacher and A. Schröder, Space adaptive finite element methods for dynamic Signorini problems., English, Comput. Mech., 44(4):481-491, 2009
[ doi | url | ]
Summary: Space adaptive techniques for dynamic Signorini problems are discussed. For discretisation, the Newmark method in time and low order finite elements in space are used. For the global discretisation error in space, an a posteriori error estimate is derived on the basis of the semi-discrete problem in mixed form. This approach relies on an auxiliary problem, which takes the form of a variational equation. An adaptive method based on the estimate is applied to improve the finite element approximation. Numerical results illustrate the performance of the presented method.

@article{Blum09CM,
  author = {Blum, Heribert and Rademacher, Andreas and Schr{\"o}der, Andreas},
  title = {{Space adaptive finite element methods for dynamic Signorini problems.}},
  language = {English},
  journal = {Comput. Mech.},
  volume = {44},
  number = {4},
  pages = {481-491},
  year = 2009,
  doi = {10.1007/s00466-009-0385-4},
  abstract = {Summary: Space adaptive techniques for dynamic Signorini problems are discussed. For discretisation, the Newmark method in time and low order finite elements in space are used. For the global discretisation error in space, an a posteriori error estimate is derived on the basis of the semi-discrete problem in mixed form. This approach relies on an auxiliary problem, which takes the form of a variational equation. An adaptive method based on the estimate is applied to improve the finite element approximation. Numerical results illustrate the performance of the presented method.},
  keywords = {dynamic Signorini problem; a posteriori error estimation; mesh refinement; finite element method},
  classmath = {*74-99 (Mechanics of deformable solids)},
  url = {http://www.springerlink.com/content/j869t1k364081830/?p=5111558c1f154257a4e040547e4bd56e&pi=8}
}
H. Blum, A. Rademacher and A. Schröder, Space adaptive finite element methods for dynamic obstacle problems., English, ETNA, Electron. Trans. Numer. Anal., 32:162-172, 2008
[ pdf | url | ]
Summary: The necessity to approximate dynamic contact problems arises in many engineering processes. Because of the local effects in the contact zone, adaptive techniques are suited to improve the finite element discretisation of such problems. In this article, the Newmark method in time and finite elements in space are used to approximate the solution numerically. A spatial error estimator is derived from the semidiscretised problem. The approach relies on an auxiliary problem, which is a variational equation. An adaptive refinement process is based on this error control. Numerical results illustrate the performance of the presented method.

@article{Blum08ETNA,
  author = {Blum, Heribert and Rademacher, Andreas and Schr{\"o}der, Andreas},
  title = {{Space adaptive finite element methods for dynamic obstacle problems.}},
  language = {English},
  journal = {ETNA, Electron. Trans. Numer. Anal.},
  volume = {32},
  pages = {162-172},
  year = 2008,
  abstract = {Summary: The necessity to approximate dynamic contact problems arises in many engineering processes. Because of the local effects in the contact zone, adaptive techniques are suited to improve the finite element discretisation of such problems. In this article, the Newmark method in time and finite elements in space are used to approximate the solution numerically. A spatial error estimator is derived from the semidiscretised problem. The approach relies on an auxiliary problem, which is a variational equation. An adaptive refinement process is based on this error control. Numerical results illustrate the performance of the presented method.},
  keywords = {{a posteriori error estimation; mesh refinement; Newmark method in time; finite elements in space}},
  classmath = {{*35L85 (Unilateral problems; variational inequalities (hyperbolic type)) 65M50 (Mesh generation and refinement (IVP of PDE)) 65M60 (Finite numerical methods (IVP of PDE))}},
  url = {http://www.emis.de/journals/ETNA/vol.32.2008/pp162-172.dir/pp162-172.html},
  pdf = {http://www.emis.de/journals/ETNA/vol.32.2008/pp162-172.dir/pp162-172.pdf},
  ps = {http://www.emis.de/journals/ETNA/vol.32.2008/pp162-172.dir/pp162-172.ps}
}
K. Weinert, H. Blum, B. Kuhlenkötter, A. Schröder, X. Zhang, C. Scheele and T. Surmann, New methods for calculating the force distribution within belt grinding processes., Production Engineering, 3:285-289, 2007
[ ]

@article{Weinert07PE,
  author = {Weinert, Klaus and Blum, Heribert and Kuhlenk{\"o}tter, Bernd and Schr{\"o}der, Andreas and Zhang, X. and Scheele, Carsten and Surmann, Tobias},
  title = {{New methods for calculating the force distribution within belt grinding processes.}},
  journal = {Production Engineering},
  volume = {3},
  pages = {285-289},
  year = 2007
}

Submitted

C. Carstensen, A. Schröder and S. Wiedemann, An Optimal Adaptive Finite Elements Method for Elastoplasticity, submitted to Numerische Mathematik, 2012
[ ]
An adaptive finite element algorithm for problems in elastoplasticity with hardening is proven to be of optimal convergence with respect to the notion of approximation classes. The results rely on the equivalence of the errors of the stresses and energies resulting from Jensen's inequality. Numerical experiments study the influence of the hardening and bulk parameters to the convergence behavior of the AFEM algorithm. This is the first optimal adaptive FEM for a variational inequality.

@unpublished{Carstensen2012NMb,
  author = {Carstensen, Carsten and Schr{\"o}der, Andreas and Wiedemann, Sebastian},
  title = {An Optimal Adaptive Finite Elements Method for Elastoplasticity},
  note = {submitted to {\em Numerische Mathematik}},
  year = 2012,
  abstract = {An adaptive finite element algorithm for problems in elastoplasticity with hardening is proven to be of optimal convergence with respect to the notion of approximation classes. The results rely on the equivalence of the errors of the stresses and energies resulting from Jensen's inequality. Numerical experiments study the influence of the hardening and bulk parameters to the convergence behavior of the AFEM algorithm. This is the first optimal adaptive FEM for a variational inequality.
  }
}
C. Carstensen, D. Peterseim and A. Schröder, The norm of a discretized gradient in H(div)* for a posteriori finite element error analysis, submitted to Numerische Mathematik, 2012
[ ]
This paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in H(div) and displacements in L2. Reliable and efficcient error estimates are introduced. One key argument in the analysis is some Fortin-operator which maps gradients in the kernel of the residual. For the Laplace, Lam´e, Stokes, and eddy current equations, this recovers old and introduces new a posteriori error estimators.

@unpublished{Carstensen2012NM,
  author = {Carstensen, Carsten and Peterseim, Daniel and Schr{\"o}der, Andreas},
  title = {The norm of a discretized gradient in H(div)* for a posteriori finite element error analysis},
  note = {submitted to {\em Numerische Mathematik}},
  year = 2012,
  abstract = {This paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in H(div) and displacements in L2. Reliable and efficcient error estimates are introduced. One key argument in the analysis is some Fortin-operator which maps gradients in the kernel of the residual. For the Laplace, Lam´e, Stokes, and eddy current equations, this recovers old and introduces new a posteriori error estimators.
  }
}
A. Schröder, H. Kleemann and H. Blum, Mixed finite element methods for two-body contact problems, submitted to Computational and Applied Mathematics, Humboldt Universität zu Berlin, Department of Mathematics, Preprint 11-02, 2011
[ pdf | url | ]
This paper presents mixed finite element methods of higher-order for two-body contact problems of linear elasticity. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The main focus is on the convergence of the scheme and on a priori estimates for the h- and p-method. For this purpose, a discrete inf-sup condition is proven which, moreover, guarantees the stability of the mixed method. Numerical results confirm the theoretical findings.

@unpublished{Schroeder2011TwoBody,
  author = {Schr{\"o}der, Andreas and Kleemann, Heiko and Blum, Heribert},
  title = {Mixed finite element methods for two-body contact problems},
  note = {submitted to Computational and Applied Mathematics, Humboldt Universit{\"a}t zu Berlin, Department of Mathematics, 
Preprint 11-02},
  year = 2011,
  abstract = {This paper presents mixed finite element methods of higher-order for two-body contact problems of linear elasticity. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The main focus is on the convergence of the scheme and on a~priori estimates for the $h$- and $p$-method. For this purpose, a discrete inf-sup condition is proven which, moreover, guarantees the stability of the mixed method. Numerical results confirm the theoretical findings.},
  url = {http://www2.mathematik.hu-berlin.de/publ/pre/2011/M-11-02.html},
  pdf = {http://www2.mathematik.hu-berlin.de/publ/pre/2011/P-11-02.pdf}
}
A. Rademacher, A. Schröder, H. Blum and H. Kleemann, Mixed FEM of higher order for time-dependent contact problems, submitted to Applied Mathematics and Computation, Ergebnisberichte des Instituts für Angewandte Mathematik, TU Dortmund, Preprint 431, 2011
[ pdf | url | ]
In this paper mixed finite element methods of higher-order for time-dependent contact problems are discussed. The mixed methods are based on resolving the contact conditions by the introduction of Lagrange multipliers. Dynamic Signorini problems with and without friction are considered involving thermomechanical and rolling contact. Rothe’s method is used to provide a suitable time and space discretization. To discretize in time, a stabilized Newmark method is applied as an adequate time stepping scheme. The space discretization relies on finite elements of higher-order. In each time step the resulting problems are solved by Uzawa‘s method or, alternatively, by methods of quadratic programming via a suitable formulation in terms of the Lagrange multipliers. Numerical results are presented towards an application in production engineering. The results illustrate the performance of the presented techniques for a variety of problem formulations.

@unpublished{Schroeder2011AMC,
  author = {Rademacher, Andreas and Schr{\"o}der, Andreas and Blum, Heribert and Kleemann, Heiko},
  title = {Mixed FEM of higher order for time-dependent contact problems},
  note = {submitted to Applied Mathematics and Computation, Ergebnisberichte des Instituts f{\"u}r Angewandte Mathematik, TU 
Dortmund, Preprint 431},
  year = 2011,
  abstract = {In this paper mixed finite element methods of higher-order for time-dependent contact problems are discussed. The mixed methods are based on resolving the contact conditions by the introduction of Lagrange multipliers. Dynamic Signorini problems with and without friction are considered involving thermomechanical and rolling contact. Rothe’s method is used to provide a suitable time and space discretization. To discretize in time, a stabilized Newmark method is applied as an adequate time stepping scheme. The space discretization relies on finite elements of higher-order. In each time step the resulting problems are solved by Uzawa‘s method or, alternatively, by methods of quadratic programming via a suitable formulation in terms of the Lagrange multipliers. Numerical results are presented towards an application in production engineering. The results illustrate the performance of the presented techniques for a variety of problem formulations.},
  url = {http://www.mathematik.tu-dortmund.de/de/forschung/publikationen/publikationen_iam.html},
  pdf = {http://www.mathematik.tu-dortmund.de/papers/RademacherSchroederBlumKleemann2011.pdf}
}

Reviewed Proceedings

L. Walczak, F. Weichert, A. Schröder, C. Landes, H. Müller and M. Wagner, Einfluss von Formvariationen auf Finite Elemente Simulationen bei muskulären Strukturen, Informatik aktuell, Bildverarbeitung für die Medizin 2010:182-186, 2010
[ url | ]

@inproceedings{Walczak10BM,
  author = {Walczak, Lars and Weichert, Frank and Schr{\"o}der, Andreas and Landes, C. and M{\"u}ller, H. and Wagner, M.},
  journal = {{Informatik aktuell}, Bildverarbeitung f{{\"u}}r die Medizin 2010},
  title = {{E}influss von {F}ormvariationen auf {F}inite {E}lemente {S}imulationen bei muskul{\"a}ren {S}trukturen},
  pages = {182-186},
  year = 2010,
  url = {http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-574/}
}
A. Schröder, Constrained approximation in hp-FEM: Unsymmetric subdivisions and multilevel hanging nodes., Lecture Notes in Computational Sciences and Engineering (ICOSAHOM Proceedings), 76:317-326, 2010
[ pdf | ]
In conform hp-finite element schemes on irregular meshes, one has to ensure the finite element functions to be continuous across edges and faces in the presence of hanging nodes. A key approach is to constrain the appropriate shape functions using so-called connectivity matrices. In this work the connectivity matrices for hierarchical tensor product shape functions are explicitly determined. In particular, the presented approach includes both unsymmetric subdivsions and multlilevel hanging nodes not using hierarchical or multi-level information of subdivions. Moreover, the problem of edge and face orientations is considered.

@inproceedings{Schroeder09ISO,
  author = {Schr{\"o}der, Andreas},
  title = {{Constrained approximation in $hp$-FEM: Unsymmetric subdivisions and multilevel hanging nodes.}},
  journal = {{Lecture Notes in Computational Sciences and Engineering (ICOSAHOM Proceedings)}},
  volume = {76},
  abstract = {In conform hp-finite element schemes on irregular meshes, one has to ensure the finite element functions to be continuous across edges and faces in the presence of hanging nodes. A key approach is to constrain the appropriate shape functions using so-called connectivity matrices. In this work the connectivity matrices for hierarchical tensor product shape functions are explicitly determined. In particular, the presented approach includes both unsymmetric subdivsions and multlilevel hanging nodes not using hierarchical or multi-level information of subdivions. Moreover, the problem of edge and face orientations is considered.},
  pages = {317-326},
  year = 2010,
  pdf = {http://www.math.hu-berlin.de/~compmath/publications/../downloads/schroeder.pdf}
}
A. Schröder, A posteriori error estimation in Mixed Finite Element Methods for Signorini's Problem., Numerical mathematics and advanced applications. Proceedings of ENUMATH 2009:801-808, 2010
[ pdf | ]
This paper presents a posteriori error estimates for Signorini’s problem which is discretized via a mixed finite element approach. The error control relies on the estimation of the discretization error of an auxiliary problem given as a variational equation. The resulting error estimates capture the discretization error of the auxiliary problem, the geometrical error and the error given by the complementary condition. The estimates are applied within adaptive finite element schemes. Numerical results confirm the applicability of the theoretical findings.

@inproceedings{Schroeder09ENU,
  author = {Schr{\"o}der, Andreas},
  title = {{A posteriori error estimation in Mixed Finite Element Methods for Signorini's Problem.}},
  journal = {Numerical mathematics and advanced applications. Proceedings of ENUMATH 2009},
  pages = {801-808},
  year = 2010,
  abstract = {This paper presents a posteriori error estimates for Signorini’s problem which is discretized via a mixed finite element approach. The error control relies on the estimation of the discretization error of an auxiliary problem given as a variational equation. The resulting error estimates capture the discretization error of the auxiliary problem, the geometrical error and the error given by the complementary condition. The estimates are applied within adaptive finite element schemes. Numerical results confirm the applicability of the theoretical findings.},
  pdf = {http://www.math.hu-berlin.de/~compmath/publications/../downloads/schroeder_enumath09.pdf}
}
M. Deichmüller, B. Denkena, K. M. de Payrebrune, M. Kröger, S. Wiedemann, A. Schröder and C. Carstensen, Determination of Static and Dynamic Deflections in Tool Grinding using a Dexel-Based Material Removal Simulation, CIRP - 2nd International Conference on Process Machine Interactions - Proceedings., 2010
[ ]

@inproceedings{Deichmueller10PMI,
  author = {Deichm{\"u}ller, M. and Denkena, B. and de Payrebrune, K. M. and Kr{\"o}ger, M. and Wiedemann, Sebastian and Schr{\"o}der, Andreas and Carstensen, Carsten},
  title = {Determination of Static and Dynamic Deflections in Tool Grinding using a Dexel-Based Material Removal Simulation},
  journal = {CIRP - 2$^{nd}$ International Conference on Process Machine Interactions - Proceedings.},
  year = 2010
}
F. Weichert, A. Schröder, C. Landes, L. Walczak, H. Müller and M. Wagner, Informatik aktuell, Netzgenerierung und Finite-Elemente-Simulation muskulärer Strukturen unter Beachtung korrespondierender histologischer Schnittpräparate, Bildverarbeitung für die Medizin 2009:192-196, 2009
[ doi | ]
In dieser Arbeit wird ein Verfahren zur Netzgenerierung und Finite-Elemente-Simulation muskulärer Strukturen vorgestellt. Eine Herausforderung liegt dabei in der Fusion der Simulationsergebnisse mit den Daten histologischer Schnittpräparate. Basierend auf segmentierten histologischen Daten wird eine Rekonstruktion der relevanten muskulären Strukturen mittels eines BCC-Tetraedernetzes initiiert. Dieses wird getriggert über einen Level-Set-Ansatz. Die Deformation des Muskels wird dann mittels eines hyperelastischen Materialgesetzes modelliert und mithilfe der Finite-Elemente-Methode simuliert. Anschließend erfolgt eine Projektion der Ergebnisse der Simulation auf die histologischen Schnittpräparate. Als Anwendungsfall wurden Daten abortiver Feten mit einer Spaltbildung im Bereich der Lippen-, Kiefer- und Gaumenregion herangezogen.

@inproceedings{Weichert09BVM,
  author = {Weichert, Frank and Schr{{\"o}}der, A. and Landes, C. and Walczak, Lars and M{{\"u}}ller, H. and Wagner, M.},
  series = {Informatik aktuell},
  journal = {{Bildverarbeitung f{{\"u}}r die Medizin 2009}},
  title = {{N}etzgenerierung und {F}inite-{E}lemente-{S}imulation muskul{{\"a}}rer {S}trukturen unter {B}eachtung korrespondierender histologischer {S}chnittpr{{\"a}}parate},
  year = 2009,
  pages = {192--196},
  abstract = {In dieser Arbeit wird ein Verfahren zur Netzgenerierung und Finite-Elemente-Simulation muskul{{\"a}}rer Strukturen vorgestellt. Eine Herausforderung liegt dabei in der Fusion der Simulationsergebnisse mit den Daten histologischer Schnittpr{{\"a}}parate. Basierend auf segmentierten histologischen Daten wird eine Rekonstruktion der relevanten muskul{{\"a}}ren Strukturen mittels eines BCC-Tetraedernetzes initiiert. Dieses wird getriggert {{\"u}}ber einen Level-Set-Ansatz. Die Deformation des Muskels wird dann mittels eines hyperelastischen Materialgesetzes modelliert und mithilfe der Finite-Elemente-Methode simuliert. Anschlie{ß}end erfolgt eine Projektion der Ergebnisse der Simulation auf die histologischen Schnittpr{\"a}parate. Als Anwendungsfall wurden Daten abortiver Feten mit einer Spaltbildung im Bereich der Lippen-, Kiefer- und Gaumenregion herangezogen.},
  doi = {10.1007/978-3-540-93860-6_39}
}
L. Walczak, F. Weichert, A. Schröder, C. Landes, H. Müller and M. Wagner, Evaluating the Impact of Shape on Finite Element Simulations in a Medical Context, Springer Berlin / Heidelberg, Lecture Notes in Computer Science, Modelling the Physiological Human:95-109, 2009
[ doi | ]
Competing concepts exist regarding surgery for instance of the cleft lip and palate to date. In order to support the surgeon to predict the possible outcome of a variety of the approaches a promising procedure are morphology-based finite element simulations at histological scale. It however can be a challenge to generate volume meshes that are applicable to the mathematical modeling of three-dimensional spatial modifications. In this study we discuss the variation of the segmentations by different anatomy experts with respect to shape, analyze the associated reconstructions by the finite element method and compare them among one another. The gist of the study is that an exact segmentation is fundamental precedent for a simulation and minor deviations in shape may arise deviations in a finite element simulation.

@incollection{Weichert09MPH,
  author = {Walczak, Lars and Weichert, Frank and Schr{{\"o}}der, A. and Landes, C. and M{{\"u}}ller, H. and Wagner, M.},
  journal = {{Lecture Notes in Computer Science, Modelling the Physiological Human}},
  title = {Evaluating the Impact of Shape on Finite Element Simulations in a Medical Context},
  publisher = {Springer Berlin / Heidelberg},
  year = 2009,
  pages = {95--109},
  abstract = {Competing concepts exist regarding surgery for instance of the cleft lip and palate to date. In order to support the surgeon to predict the possible outcome of a variety of the approaches a promising procedure are morphology-based finite element simulations at histological scale. It however can be a challenge to generate volume meshes that are applicable to the mathematical modeling of three-dimensional spatial modifications. In this study we discuss the variation of the segmentations by different anatomy experts with respect to shape, analyze the associated reconstructions by the finite element method and compare them among one another. The gist of the study is that an exact segmentation is fundamental precedent for a simulation and minor deviations in shape may arise deviations in a finite element simulation.},
  doi = {10.1007/978-3-642-10470-1_9}
}
A. Schröder, Goal-oriented error control in h- and hp-adaptive FEM for Signorini's problem, Computer Methods in Mechanics Proceedings 2009, 409-410:409-410, 2009
[ ]

@inproceedings{Schroeder09CMM,
  author = {Schr{\"o}der, Andreas},
  title = {{Goal-oriented error control in $h$- and $hp$-adaptive FEM for Signorini's problem}},
  journal = {Computer Methods in Mechanics Proceedings 2009, 409-410},
  pages = {409-410},
  year = 2009
}
A. Schröder and H. Blum, Projective SOR-Procedures for Signorini Problems, AIP, English, AIP Conference Proceedings, 1048(1):474-477, 2008
[ doi | url | ]
Summary: We present a successive overrelaxation (SOR)-procedure with projection for solving quadratic optimization problems resulting from the Signorini problems in linear elasticity. The procedure exploits the sparsity structure of problems given by, e.g., finite element discretizations. A convergence result of the procedure is given, and numerical results are presented in the context of contact problems. Especially, accelerated variants are discussed based on conjugate gradient-like methods and cascadic multigrid approaches.

@inproceedings{Schroeder08NAAM,
  language = {English},
  author = {Schr{\"o}der, Andreas and Blum, Heribert},
  editor = {Simos, T. E. and Psihoyios, G. and Tsitouras, Ch.},
  collaboration = {},
  title = {{Projective SOR-Procedures for Signorini Problems}},
  publisher = {AIP},
  year = {2008},
  journal = {AIP Conference Proceedings},
  volume = {1048},
  number = {1},
  pages = {474-477},
  abstract = {Summary: We present a successive overrelaxation (SOR)-procedure with projection for solving quadratic optimization problems resulting from the Signorini problems in linear elasticity. The procedure exploits the sparsity structure of problems given by, e.g., finite element discretizations. A convergence result of the procedure is given, and numerical results are presented in the context of contact problems. Especially, accelerated variants are discussed based on conjugate gradient-like methods and cascadic multigrid approaches.},
  keywords = {numerical examples; finite element; convergence; numerical results; successive overrelaxation procedure; conjugate gradient methods; cascadic multigrid; contact problem; signorini problem; quadralic optimization},
  classmath = {*65K05 (Mathematical programming (numerical methods)) 90C20 (Quadratic programming) 74M15 (Contact) 74S05 (Finite element methods)},
  url = {http://link.aip.org/link/?APC/1048/474/1},
  doi = {10.1063/1.2990965}
}
A. Schröder, Numerical mathematics and advanced applications. Proceedings of ENUMATH 2007, Constraints coefficients in hp-FEM., Springer Berlin / Heidelberg, English:183-190, 2008
[ pdf | ]
Abstract Continuity requirements on irregular meshes enforce a proper constraint of the degrees of freedom that correspond to hanging nodes, edges or faces. This is achieved by using so-called constraints coefficients which are obtained from the appropriate coupling of shape functions. In this note, a general framework for determining the constraints coefficients of tensor product shape functions is presented and its application to shape functions using integrated Legendre or Gauss-Lobatto polynomials. The constraints coefficients in the one-dimensional case are determined via recurrence relations. The constraints coefficients in the multi-dimensional case are obtained as products of these coefficients. The coefficients are available for arbitrary patterns of subdivisions.

@inproceedings{Schroeder08ENU,
  author = {Schr{\"o}der, Andreas},
  title = {{Constraints coefficients in $hp$-FEM.}},
  language = {English},
  booktitle = {Numerical mathematics and advanced applications. Proceedings of ENUMATH 2007},
  editor = {Kunisch, K.},
  pages = {183-190},
  year = 2008,
  abstract = {Abstract Continuity requirements on irregular meshes enforce a proper constraint of the degrees of freedom that correspond to hanging nodes, edges or faces. This is achieved by using so-called constraints coefficients which are obtained from the appropriate coupling of shape functions.
In this note, a general framework for determining the constraints coefficients of tensor product shape functions is presented and its application to shape functions using integrated Legendre or Gauss-Lobatto polynomials. The constraints coefficients in the one-dimensional case are determined via recurrence relations. The constraints coefficients in the multi-dimensional case are obtained as products of these coefficients. The coefficients are available for arbitrary patterns of subdivisions.},
  publisher = {Springer Berlin / Heidelberg},
  pdf = {http://www.math.hu-berlin.de/~compmath/publications/../downloads/schroeder_a_contrib.pdf}
}
B. Denkena, M. Deichmüller, M. Kröger, K. M. Popp, C. Carstensen, A. Schröder and S. Wiedemann, Geometrical analysis of the complex contact area for modeling the local distribution of process forces in tool grinding, CIRP - 1st International Conference on Process Machine Interactions - Proceedings.:289-298, 2008
[ ]

@inproceedings{Denkena08PMI,
  author = {Denkena, B. and Deichm{\"u}ller, M. and Kr{\"o}ger, M. and Popp, K.~M. and Carstensen, Carsten and Schr{\"o}der, Andreas and Wiedemann, Sebastian},
  title = {Geometrical analysis of the complex contact area for modeling the local distribution of process forces in tool grinding},
  howpublished = {CIRP - 1$^{st}$ International Conference on Process Machine Interactions - Proceedings.},
  pages = {289-298},
  year = 2008
}
D. Biermann, H. Blum, T. Jansen, A. Rademacher, A. Scheidler, A. Schröder and K. Weinert, Space Adaptive Finite Element Methods for Dynamic Signorini Problems in the Simumation of the NC-Shape Grinding Process., 1st International Conference on Process Machine Interactions - Proceedings.:309-316, 2008
[ ]

@inproceedings{Biermann08PMI,
  author = {Biermann, Dirk and Blum, Heribert and Jansen, T. and Rademacher, Andreas and Scheidler, A. and Schr{\"o}der, Andreas and Weinert, Klaus},
  title = {{Space Adaptive Finite Element Methods for Dynamic Signorini Problems in the Simumation of the NC-Shape Grinding Process.}},
  howpublished = {1$^{st}$ International Conference on Process Machine Interactions - Proceedings.},
  pages = {309-316},
  year = 2008
}
H. Blum, A. Schröder and F. T. Suttmeier, A posteriori estimates for FE-solutions of variational inequalities., Brezzi, Franco (ed.) et al., Numerical mathematics and advanced applications. Proceedings of ENUMATH 2001, the 4th European conference, Ischia, July 2001. Berlin: Springer. 669-680 (2003)., 2003
[ ]

@inproceedings{Blum03ENU,
  author = {Blum, Heribert and Schr{\"o}der, Andreas and Suttmeier, Franz-Theo},
  title = {{A posteriori estimates for FE-solutions of variational inequalities.}},
  howpublished = {Brezzi, Franco (ed.) et al., Numerical mathematics and advanced applications. Proceedings of ENUMATH 2001, the 4th European conference, Ischia, July 2001. Berlin: Springer. 669-680 (2003).},
  year = 2003
}

Proceedings

A. Schröder, Mixed FEM for a frictional contact problem, PAMM, 11:7-10, 2011
[ doi | pdf | ]
This paper presents mixed finite element methods of higher-order for an idealized frictional contact problem in linear elasticity. The approach relies on a saddle point formulation where the frictional contact condition is captured by a Lagrange multiplier. The convergence of the mixed scheme is proven and some a priori estimates for the h- and p-method are derived. Furthermore, a posteriori error estimates are presented which rely on the estimation of the discretization error of an auxiliary problem and some further terms capturing the error in the friction and complementary conditions. Numerical results confirm the applicability of the a posteriori error estimates within h- and hp-adaptive schemes.

@misc{Schroeder2011PAMM,
  author = {Schr{\"o}der, Andreas},
  title = {Mixed FEM for a frictional contact problem},
  journal = {PAMM},
  volume = {11},
  issue = {1},
  pages = {7-10},
  year = 2011,
  abstract = {This paper presents mixed finite element methods of higher-order for an idealized frictional contact problem in linear elasticity. The approach relies on a saddle point formulation where the frictional contact condition is captured by a Lagrange multiplier. The convergence of the mixed scheme is proven and some a~priori estimates for the $h$- and $p$-method are derived. Furthermore, a~posteriori error estimates are presented which rely on the estimation of the discretization error of an auxiliary problem and some further terms capturing the error in the friction and complementary conditions. Numerical results confirm the applicability of the a~posteriori error estimates within $h$- and $hp$-adaptive schemes.},
  pdf = {http://www2.mathematik.hu-berlin.de/publ/pre/2011/P-11-14.pdf}
  doi = {10.1002/pamm.201110003}
}
A. Schröder, hp-adaptive finite element methods for variational inequalities, PAMM, 8(1):10053-10056, 2009
[ pdf | url | ]
In this work, we combine an hp-adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction. Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp-adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence.

@misc{Schroeder09PAMM,
  author = {Schr{\"o}der, Andreas},
  title = {{$hp$-adaptive finite element methods for variational inequalities}},
  journal = {PAMM},
  volume = {8},
  number = {1},
  pages = {10053-10056},
  year = 2009,
  abstract = {In this work, we combine an $hp$-adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction. Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the $hp$-adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence.},
  url = {http://www3.interscience.wiley.com/journal/122217787/abstract?CRETRY=1&SRETRY=0},
  pdf = {http://www3.interscience.wiley.com/cgi-bin/fulltext/122217787/PDFSTART}
}
A. Byfut, A. Schröder and C. Carstensen, hp-Adaptive Extended Finite Element Method., Proceedings of the International Conference on Extended Finite Element Methods, Aachen, 31-34, 2009., 2009
[ ]

@misc{Byfut09ICXFEM,
  author = {Byfut, Andreas and Schr{\"o}der, Andreas and Carstensen, Carsten},
  title = {{$hp$-Adaptive Extended Finite Element Method.}},
  howpublished = {Proceedings of the International Conference on Extended Finite Element Methods, Aachen, 31-34, 2009.},
  year = 2009
}
A. Byfut, T. Backers, A. Schröder and C. Carstensen, Appliance of the Extended Finite Element Method in Geomechanics., Proceedings of the 71st EAGE Conference & Exhibition, Amsterdam, 2009., 2009
[ url | ]

@misc{Byfut09CE,
  author = {Byfut, Andreas and Backers, T. and Schr{\"o}der, Andreas and Carstensen, Carsten},
  title = {{Appliance of the Extended Finite Element Method in Geomechanics.}},
  howpublished = {Proceedings of the 71st EAGE Conference \& Exhibition, Amsterdam, 2009.},
  year = 2009,
  url = {http://www.earthdoc.org/detail.php?pubid=24021}
}
M. Cabaravdic, K. Kneupner, B. Kuhlenkötter and A. Schröder, Force-distribution model by the belt grinding with elastic contact wheel., In 8th. International Research/Expert Conference - Trends in the Development of Machinery and Associated Technology, 2004., 2004
[ ]

@misc{Cabaravdic04IREC,
  author = {Cabaravdic, Malik and Kneupner, Klaus and Kuhlenk{\"o}tter, Bernd and Schr{\"o}der, Andreas},
  title = {{Force-distribution model by the belt grinding with elastic contact wheel.}},
  howpublished = {In {\em 8th. International Research/Expert Conference - Trends in the Development of Machinery and Associated Technology, 2004.}},
  year = 2004
}
H. Blum, A. Schröder and F. T. Suttmeier, A posteriori error bounds for finite element schemes for a model friction problem., In Simulation Aided Offline Process Design and Optimization in Manufacturing Sculptured Surfaces, 39-47, 2003., 2003
[ ]

@misc{Blum03FGFFF,
  author = {Blum, Heribert and Schr{\"o}der, Andreas and Suttmeier, Franz-Theo},
  title = {{A posteriori error bounds for finite element schemes for a model friction problem.}},
  howpublished = {In {\em Simulation Aided Offline Process Design and Optimization in Manufacturing Sculptured Surfaces}, 39-47, 2003.},
  year = 2003
}

Preprints

A. Rademacher and A. Schröder, Dual weighted residual error control for frictional contact problems., Humboldt Universität zu Berlin, Institut für Mathematik, Preprint 12-07, 2012
[ pdf | ]
In this paper goal-oriented error control based on dual weighted residual error estima- tions (DWR) is applied to frictional contact problems. A mixed formulation of the contact problem is used to derive a discretization. It relies on the introduction of Lagrange multipliers to capture the frictional contact conditions. The discretization error is estimated in terms of functionals (the quan- tities of interest) which are evaluated in the displacement field as well as the Lagrange multipliers. Numerical experiments confirm the applicability of the estimates within adaptive schemes.

@techreport{Rademacher12,
  author = {Rademacher, Andreas and Schr{\"o}der, Andreas},
  title = {Dual weighted residual error control for frictional contact problems.},
  note = {Humboldt Universit{\"a}t zu Berlin, Institut f{\"u}r Mathematik, Preprint 12-07}}
  pdf = { http://www2.mathematik.hu-berlin.de/publ/pre/2012/P-12-07.pdf},
  year = 2012,
  abstract = { In this paper goal-oriented error control based on dual weighted residual error estima-
tions (DWR) is applied to frictional contact problems. A mixed formulation of the contact problem
is used to derive a discretization. It relies on the introduction of Lagrange multipliers to capture the
frictional contact conditions. The discretization error is estimated in terms of functionals (the quan-
tities of interest) which are evaluated in the displacement field as well as the Lagrange multipliers.
Numerical experiments confirm the applicability of the estimates within adaptive schemes.}
A. Schröder, A. Byfut, R. Joliet and T. Surmann, hp-finite element simulation of thermoelastic deformation in NC-milling, Humboldt Universität zu Berlin, Institut für Mathematik, Preprint 12-06, 2012
[ pdf | ]
Abstract This paper presents an hp-finite element approach for the simulation of heat diffusion and thermoelastic deformation in the NC milling process. Varying complex shapes resulting from the material removal are taken into account via a removal-dependent mesh refinement that relies on the isotropic bisection of hexahedrons intersecting the geometry of the milling tool in contact. To represent the cutting surface more accurately, the hexahedrons at the cutting surface are further subdivided into tetrahedrons, pyramids and prisms. To avoid the involved application of higher-order shape funtions on these shapes, and to enable the exclusive use of tensor product shape functions on hexahedrons, a fictitious domain approach is used. Numerical experiments show the applicability of the proposed method.

@techreport{Schroeder12,
  author = {Schr{\"o}der, Andreas and Byfut, Andreas and Joliet, Raffael and Surmann, Tobias},
  title = {{$hp$-finite element simulation of thermoelastic deformation in NC-milling}},
  howpublished = {Humboldt Universit{\"a}t zu Berlin, Institut f{\"u}r Mathematik, Preprint 12-06}
  year={2012},
  abstract = {Abstract This paper presents an hp-finite element approach for the simulation of heat diffusion and thermoelastic deformation in the NC milling process. Varying complex shapes resulting from the material removal are taken into account via a removal-dependent mesh refinement that relies on the isotropic bisection of hexahedrons intersecting the geometry of the milling tool in contact. To represent the cutting surface more accurately, the hexahedrons at the cutting surface are further subdivided into tetrahedrons, pyramids and prisms. To avoid the involved application of higher-order shape funtions on these shapes, and to enable the exclusive use of tensor product shape functions on hexahedrons, a fictitious domain approach is used. Numerical experiments show the applicability of the proposed method.}
  pdf = {http://www.math.hu-berlin.de/~compmath/publications/../downloads/schroeder12.pdf}
}

Dissertation

A. Schröder, Fehlerkontrollierte adaptive h- und hp-Finite-Elemente-Methoden für Kontaktprobleme mit Anwendungen in der Fertigungstechnik., German, 2006
[ pdf | ]

@phdthesis{Schroeder06BMS,
  author = {Schr{\"o}der, Andreas},
  title = {{Fehlerkontrollierte adaptive $h$- und $hp$-Finite-Elemente-Methoden f{\"u}r Kontaktprobleme mit Anwendungen in der Fertigungstechnik.}},
  language = {German},
  school = {Dortmund: Univ. Dortmund, Fachbereich Mathematik (Diss.). Bayreuther Math. Schr. 78, xviii, 216~p.},
  year = 2006,
  pdf = {https://eldorado.uni-dortmund.de/bitstream/2003/22487/1/dissertation.pdf}
}