Applied Mathematics
TU Dortmund University

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• Dual weighted residual error estimation for the finite cell method
Di Stolfo, P. and Rademacher, A. and Schröder, A.
Journal of Numerical Mathematics 27 (2019)
The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples. © 2019 Walter de Gruyter GmbH, Berlin/Boston.
 view abstract 10.1515/jnma-2017-0103
• Goal-oriented a posteriori error estimates in nearly incompressible linear elasticity
Lecture Notes in Computational Science and Engineering 126 (2019)
In this article, we consider linear elastic problems, where Poisson’s ratio is close to 0.5 leading to nearly incompressible material behavior. The use of standard linear or d-linear finite elements involves locking phenomena in the considered problem type. One way to overcome this difficulties is given by selective reduced integration. However, the discrete problem differs from the continuous one using this approach. This fact has especially to be taken into account, when deriving a posteriori error estimates. Here, we present goal-oriented estimates based on the dual weighted residual method using only the primal residual due to the linear problem considered. The major challenge is given by the construction of an appropriate numerical approximation of the error identity. Numerical results substantiate the accuracy of the presented estimator and the efficiency of the adaptive method based on it. © Springer Nature Switzerland AG 2019.
 view abstract 10.1007/978-3-319-96415-7_35
• Mesh and model adaptivity for frictional contact problems
Numerische Mathematik 142 (2019)
The article focuses on adaptive finite element methods for frictional contact problems. The approach is based on a reformulation of the mixed form of the underlying Signorini problem with friction as a nonlinear variational equation using nonlinear complimentarity functions. The usual dual weighted residual framework for a posteriori error estimation is applied. However, we have to take into account the nonsmoothness of the problem formulation. Error identities for measuring the discretization as well as the model error with respect to a model hierarchy of friction laws are derived and a method for the numerical evaluation of them is proposed. The estimates are utilized in an adaptive framework, which balances the discretization and the model error. Several numerical examples substantiate the accuracy of the proposed estimates and the efficiency of the adaptive method. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
 view abstract 10.1007/s00211-019-01044-8
• Adaptive optimal control of Signorini’s problem
Computational Optimization and Applications 70 (2018)
In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini’s problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton’s method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini’s contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini’s contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini’s problem in two and three dimensions. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
 view abstract 10.1007/s10589-018-9982-5
• NCP function-based dual weighted residual error estimators for Signorini's problem
SIAM Journal on Scientific Computing 38 (2016)
In this paper, we consider goal-oriented adaptive finite element methods for Signorini's problem. The basis is a mixed formulation, which is reformulated as nonlinear variational equality using a nonlinear complementarity function. For a general discretization, we derive error identities w.r.t. a possible nonlinear quantity of interest in the displacement as well as in the contact forces, which are included as Lagrange multiplier, using the dual weighted residual method. Afterwards, a numerical approximation of the error identities is introduced. We exemplify the results for a low order mixed discretization of Signorini's problem. The theorectical findings and the numerical approximation scheme are finally substantiated by some numerical examples. © 2016 Society for Industrial and Applied Mathematics.
 view abstract 10.1137/15M1033873
• Semi-smooth Newton methods for mixed FEM discretizations of higher-order for frictional, elasto-plastic two-body contact problems
Blum, H. and Frohne, H. and Frohne, J. and Rademacher, A.
Computer Methods in Applied Mechanics and Engineering 309 (2016)
In this article a semi-smooth Newton method for frictional two-body contact problems and a solution algorithm for the resulting sequence of linear systems are presented. It is based on a mixed variational formulation of the problem and a discretization by finite elements of higher-order. General friction laws depending on the normal stresses and elasto-plastic material behavior with linear isotropic hardening are considered. Numerical results show the efficiency of the presented algorithm. © 2016 Elsevier B.V.
 view abstract 10.1016/j.cma.2016.06.004
• Simulation of sheet-bulk metal forming processes with simufact.forming using user-subroutines
Beese, S. and Beyer, F. and Blum, H. and Isik, K. and Kumor, D. and Löhnert, S. and Rademacher, A. and Tekkaya, A.E. and Willner, K. and Wriggers, P. and Zeller, S.
AIP Conference Proceedings 1769 (2016)
In this article, we consider the simulation of sheet-bulk metal forming processes, which makes high demands on the underlying models and on the simulation software. We present our approach to incorporate new modelling approaches from various fields in a commercial simulation software, in our case Simufact.forming. Here, we discuss material, damage, and friction models as well as model adaptive techniques. © 2016 Author(s).
 view abstract 10.1063/1.4963457
• Adaptive optimal control of the obstacle problem
Meyer, C. and Rademacher, A. and Wollner, W.
SIAM Journal on Scientific Computing 37 (2015)
This article is concerned with the derivation of a posteriori error estimates for optimization problems subject to an obstacle problem. To circumvent the nondifferentiability inherent to this type of problem, we introduce a sequence of penalized but differentiable problems. We show differentiability of the central path and derive separate a posteriori dual weighted residual estimates for the errors due to penalization, discretization, and iterative solution of the discrete problems. The effectivity of the derived estimates and of the adaptive algorithm is demonstrated on two numerical examples. © 2015 Society for Industrial and Applied Mathematics.
 view abstract 10.1137/140975863
• Dual Weighted Residual Error Control for Frictional Contact Problems
Computational Methods in Applied Mathematics 15 (2015)
In this paper goal-oriented error control based on the dual weighted residual error method (DWR) is applied to frictional contact problems. The derivation of DWR error controls is done for arbitrary discretization schemes via the introduction of some discrete Lagrange multipliers describing the residual of the discretization. The discrete Lagrange multipliers may be provided by a reconstruction in a post-processing step or by a discretization of a mixed formulation in which they are directly available. The error controls are defined for user-defined functionals (the quantities of interest) which measure the error of the displacement field as well as the normal and tangential contact forces. Numerical experiments confirm the applicability of the estimates within adaptive schemes. © 2015 by De Gruyter.
 view abstract 10.1515/cmam-2015-0014
• Experimental and simulative investigations of tribology in sheet-bulk metal forming
Beyer, F. and Blum, H. and Kumor, D. and Rademacher, A. and Willner, K. and Schneider, T.
Key Engineering Materials 639 (2015)
Friction has a considerable influence in metal forming both in economic and technical terms. This is especially true for sheet-bulk metal forming (SBMF). The contact pressure that occurs here can be low making Coulomb's friction law advisable, but also very high so that Tresca's friction law is preferable. By means of an elasto-plastic half-space model rough surfaces have been investigated, which are deformed in such contact states. The elasto-plastic half-space model has been verified and calibrated experimentally. The result is the development of a constitutive friction law, which can reproduce the frictional interactions for both low and high contact pressures. In addition, the law gives conclusion regarding plastic smoothening of rough surfaces. The law is implemented in the framework of the finite element method (FEM). However, compared to usual friction relations the tribological interplay presented here comes with the disadvantage of rising numerical effort. In order to minimise this drawback, a model adaptive finite element simulation is performed additionally. In this approach, contact regions are identified, where a conventional friction law is applicable, where the newly developed constitutive friction law should be used, or where frictional effects are negligible. The corresponding goal-oriented indicators are derived based on the "dual-weighted-residual" (DWR) method taking into account both the model and the discretisation error. This leads to an efficient simulation that applies the necessary friction law in dependence of contact complexity. © 2015 Trans Tech Publications, Switzerland.
 view abstract 10.4028/www.scientific.net/KEM.639.283
• Simulation of MQL deep hole drilling for predicting thermally induced workpiece deformations
Biermann, D. and Blum, H. and Frohne, J. and Iovkov, I. and Rademacher, A. and Rosin, K.
Procedia CIRP 31 (2015)
The resulting thermomechanical load on the workpiece in deep hole drilling operations using minimum quantity lubrication (MQL) induces a strong in-process deflection of the machined component and can cause an insufficient accuracy of the produced hole. Also subsequent machining operations can be affected by the thermoelastic component of this deformation, which remains within the workpiece after the drilling process. Due to the comparatively long main time of typical deep hole drilling operations the thermomechanical simulation of commonly complex machined parts is challenging. In this paper, a fast finite-element approach using massive parallel solution methods is presented and validated for different wall thickness situations. © 2015 The Authors. Published by Elsevier B.V.
 view abstract 10.1016/j.procir.2015.03.038
• Mixed FEM of higher-order for time-dependent contact problems
Rademacher, A. and Schröder, A. and Blum, H. and Kleemann, H.
Applied Mathematics and Computation 233 (2014)
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. © 2014 Published by Elsevier Ltd.
 view abstract 10.1016/j.amc.2014.01.063
• Simulation of grinding processes using finite element analysis and geometric simulation of individual grains
Siebrecht, T. and Biermann, D. and Ludwig, H. and Rausch, S. and Kersting, P. and Blum, H. and Rademacher, A.
Production Engineering 8 (2014)
The wear-resistance of sheet metal forming tools can be increased by thermally sprayed coatings. However, without further treatment, the high roughness of the coatings leads to poor qualities of the deep drawn sheet surfaces. In order to increase the surface quality of deep drawing tools, grinding on machining centers is a suitable solution. Due to the varying engagement situations of the grinding tools on free-formed surfaces, the process forces vary as well, resulting in inaccuracies of the ground surface shape. The grinding process can be optimized by means of a simulative prediction of the occurring forces. In this paper, a geometric-kinematic simulation coupled with a finite element analysis is presented. Considering the influence of individual grains, an additional approximation to the resulting topography of the ground surface is possible. By using constructive solid geometry and dexel modeling techniques, multiple grains can be simulated with the geometric-kinematic approach simultaneously. The process forces are predicted with the finite element method based on an elasto-plastic material model. Single grain engagement experiments were conducted to validate the simulation results. © 2014 German Academic Society for Production Engineering (WGP).
 view abstract 10.1007/s11740-013-0524-9
• Stabilization techniques and a posteriori error estimates for the obstacle problem
Biermann, D. and Iovkov, I. and Blum, H. and Rademacher, A. and Klein, N. and Suttmeier, F.-T.
Applied Mathematical Sciences 7 (2013)
This work deals with a posteriori error estimates for the obstacle problem. Deriving an estimator on the basis of the variational inequal- ity with respect to the primal variable, an inconsistent one is obtained. To achieve consistency, this problem is treated by a Lagrange formalism, which transfers the variational inequality into a saddle point problem. Different techniques to ensure the stability of the discretization and to solve the discrete problems by iterative solvers are studied and com- pared. Numerical tests confirm our results of consistent a posteriori error estimation. © 2013 Dirk Biermann et al.
 view abstract 10.12988/ams.2013.39504
• Goal oriented error control for frictional contact problems in metal forming
Blum, H. and Rademacher, A. and Schröder, A.
Key Engineering Materials 504-506 (2012)
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.© (2012) Trans Tech Publications.
 view abstract 10.4028/www.scientific.net/KEM.504-506.987
• Thermal aspects in deep hole drilling of aluminium cast alloy using twist drills and MQL
Biermann, D. and Iovkov, I. and Blum, H. and Rademacher, A. and Taebi, K. and Suttmeier, F.T. and Klein, N.
Procedia CIRP 3 (2012)
The deep hole drilling process with solid carbide twist drills is an efficient alternative to the classic single-lip deep hole drilling, due to the generally higher feed rates possible and the consequently higher productivity. Furthermore the minimum quantity lubrication (MQL) can be applied, in order to reduce the production costs and implement an environmentally friendly process. Because of the significantly reduced cooling performance when using MQL, a higher heat loading results for the tool and the workpiece. This paper presents the investigations of the temperature distribution in the workpiece and the heat balance of the deep hole drilling process. © 2012 The Authors.
 view abstract 10.1016/j.procir.2012.07.043
• Goal-oriented error control in adaptive mixed FEM for Signorini's Problem
Computer Methods in Applied Mechanics and Engineering 200 (2011)
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. © 2010 Elsevier B.V.
 view abstract 10.1016/j.cma.2010.08.015
• Adaptive Finite Element Discretisation of the Spindle Grinding Wheel System
Progress in Industrial Mathematics at Ecmi 2008 15 (2010)
In the simulation of the NC-shape grinding process, a finite element model of the grinding machine is included. To enhance the accuracy and efficiency of the finite element computation, a posteriori error estimation and resulting adaptive mesh refinement techniques are used. In this note, a dual weighted a posteriori error estimate for a linear second order hyperbolic model problem is derived. Numerical results illustrate the performance of the presented approach.
 view abstract 10.1007/978-3-642-12110-4_78

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