Heribert Blum

A domain splitting algorithm for parabolic problems

In the parallel implementation of solution methods for parabolic problems one has to find a proper balance between the parallel efficiency of a fully explicit scheme and the need for stability and accuracy which requires some degree of implicitness. As a compromise a domain splitting scheme is proposed which is locally implicit on slightly overlapping subdomains but propagates the corresponding boundary data by a simple explicit process. The analysis of this algorithm shows that it has satisfactory stability and approximation properties and can be effectively parallelized. These theoretical results are confirmed by numerical tests on a transputer system.ZusammenfassungDie Implementierung von Lösungs-methoden für parabolische Probleme erfordert eine ausreichende Balance zwischen der parallelen Effizienz voll-expliziter Schemata und der Notwendigkeit von Stabilität und Genauigkeit, welche einen gewissen Grad an Implizitheit bedingt. Als ein Kompromiß wird ein Gebietszerlegungsverfahren vorgeschlagen, welches lokal implizit ist auf leicht überlappenden Teilgebieten, die lokalen Randdaten aber durch einen einfachen expliziten Prozeß fortpflanzt. Die Analyse dieses Algorithmus zeigt, daß er zufriedenstellende Stabilitäts- und Approximationseigenschaften besitzt und effektiv parallelisiert werden kann. Diese theoretischen Resultate werden bestätigt durch numerische Tests auf einem Transputer-System.

Simulation based optimization of the NC-shape grinding process with toroid grinding wheels

For achieving high material removal rates while grinding free formed surfaces, shape grinding with toroid grinding wheels is favored. The material removal is carried out line by line. The contact area between grinding wheel and workpiece is therefore complex and varying. Without detailed knowledge about the contact area, which is influenced by many factors, the shape grinding process can only be performed sub-optimally. To improve this flexible production process and in order to ensure a suitable process strategy a simulation-tool is being developed. The simulation comprises a geometric-kinematic process simulation and a finite elements simulation. This paper presents basic parts of the investigation, modelling and simulation of the NC-shape grinding process with toroid grinding wheels.

Weighted error estimates for finite element solutions of variational inequalities

Abstract In this note the studies begun in Blum and Suttmeier (1999) on adaptive finite element discretisations for nonlinear problems described by variational inequalities are continued. Similar to the concept proposed, e.g., in Becker and Rannacher (1996) for variational equalities, weighted a posteriori estimates for controlling arbitrary functionals of the discretisation error are constructed by using a duality argument. Numerical results for the obstacle problem demonstrate the derived error bounds to be reliable and, used for an adaptive grid refinement strategy, to produce economical meshes.

Angepasste Simulationstechnik zur Analyse NC-gesteuerter Formschleifprozesse

Kurzfassung Simulationen können als Hilfsmittel zur Optimierung von Fertigungsprozessen eingesetzt werden. Bei der Fertigung frei geformter Werkstücke durch Schleifen, unter Einsatz von Torusschleifscheiben, sind genaue Kenntnisse über den kontinuierlich variierenden Eingriff der Schleifscheibe in das Werkstück unentbehrlich. Die folgenden Ausführungen beschreiben die Vorgehensweise bei der Simulation des NC-Formschleifprozesses auf Basis einer geometrisch-kinematischen Simulation und einer Finite-Elemente-Simulation, deren Genauigkeit durch den Einsatz adaptiver Finite-Elemente-Methoden gesteigert wird.

Simulation of grinding processes using finite element analysis and geometric simulation of individual grains

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.

Grinding of Arc-Sprayed Tungsten Carbide Coatings on Machining Centers - Process Configuration and Simulation

This paper describes the special demands placed on the grinding of arc-sprayed WC-Fe coatings on a conventional machining center. Basic process configuration, experimental results, measurement methods and an approach for a hybrid simulation system are presented.

Mixed finite element methods for two-body contact problems

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 the p -method. For this purpose, a discrete inf-sup condition is proven which guarantees the stability of the mixed method. Numerical results confirm the theoretical findings.

New methods for calculating the force distribution within belt grinding processes

In order to simulate belt grinding processes (e.g. for process planning or path planning) one usually needs information about the contact zone and contact forces. Typically, an unacceptable computational effort is required for good simulation results, since these contact problems are usually of a nonlinear nature. In this paper, the application of support vector machines (SVM) is presented. The SVM is a learning machine that aims at finding a function that optimally fits given observations. The main advantage of SVM is its fast evaluation during simulation. However, a single training phase with an extensive amount of observation data has to be done once before the simulation can take place. From a practical point of view, it is very often not feasible to sample these observation data by experiments. At this point special Finite element methods for contact problems can be applied very efficiently. In order to obtain as accurate as possible training data, an adaptive finite element method for contact problems has been developed.

Experimental and Simulative Investigations of Tribology in Sheet-Bulk Metal Forming

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. 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.

Projective SOR‐Procedures for Signorini Problems

In this work, we present a SOR‐procedure with projection for solving quadratic optimization problems resulting from 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 CG‐like methods and cascadic multigrid approaches.