The principle of indirect elimination

The principle of indirect elimination states that an algorithm for solving discretized differential equations can be used to identify its own bad-converging modes. When the number of bad-converging modes of the algorithm is not too large, the modes thus identified can be used to strongly improve the convergence. The method presented here is applicable to any standard algorithm like Conjugate Gradient, relaxation or multigrid. An example from theoretical physics, the Dirac equation in the presence of almost-zero modes arising from instantons, is studied. Using the principle, bad-converging modes are removed efficiently. Applied locally, the principle is one of the main ingredients of the Iteratively Smooting Unigrid algorithm.Comment: 16 pages, LaTeX-style espart (elsevier preprint style). Three .eps-figures are now added with the figure command

Localization in Lattice Gauge Theory and a New Multigrid Method

We show numerically that the lowest eigenmodes of the 2-dimensional Laplace-operator with SU(2) gauge couplings are strongly localized. A connection is drawn to the Anderson-Localization problem. A new Multigrid algorithm, capable to deal with these modes, shows no critical slowing down for this problem.Comment: LATeX style, 11 pages (plus 4 figure pages). Figure pages are available as uuencoded ps-file via anonymous ftp from x4u2.desy.de, get pub/outgoing/baeker/heplat.uu. DESY-preprint 94-07

Johnson-Cook parameter identification from machining simulations using an inverse method

The Johnson-Cook model is a material model which has been widely used for simulating the chip formation processes. It is a simple 5 parameter material model which predicts the flow stress at large strains, strain-rates and at high temperatures. These parameters are usually identified by determining the flow stress curves experimentally, and then using curve fitting techniques to find the optimal parameters to describe the material behaviour. However the state-of-the-art experimental methods can only rely on data obtained from strains of up to 50% and strain-rates of the order of 103 per second, whereas in machining processes strains of more than 200% are reached at strain-rates of the order of 106 or more. Therefore, the parameters obtained at much milder conditions have limited applicability when simulating machining. In this paper an inverse method of material parameter identification from machining simulations is described. It is shown that by using the observables of a machining process such as the chip shape and cutting forces, the underlying material parameters can be identified. In order to achieve this, a finite element model of the machining process is created and simulation is carried out using a known standard parameter set from literature. The objective of the inverse method is to reidentify this set by using the chip shape and cutting forces. An error function is created using the non-overlap area of the chip shapes and the difference in the cutting forces. The Levenberg-Marquardt algorithm is used to minimise the error function. It has been shown before that multiple sets of Johnson-Cook parameter sets exist which might give rise to indistinguishable chip shapes and cutting forces. In order to identify the parameter set uniquely, simulations are performed at widely varying cutting conditions such as differing rake angles, cutting speeds and non-adiabatic conditions. Thus, material parameters which represent the material behaviour over a wide range can be identified

Crack propagation in thermal barrier coating systems

Thermal barrier coating systems are used on top of highly stressed components in gas turbines to protect the nickel-based substrates. A well-established thermal barrier coating system consist of the bond-coat (BC) and the thermal barrier coating (TBC). A third layer is grown during service: the thermally grown oxide (TGO) between the BC and the TBC. The coatings fail in service because of different failure mechanism which are not fully understood yet because of the complex interaction of different phenomena (e.g., creep, sintering, thermal differential expansion, diffusion, and oxidation). Therefore, a simplified model system is presented which consists of a FeCrAlY bulk material and a TBC applied as a top coat by atmospheric plasma spraying. This simplified model system is used to study the influence of BC and TGO creep and the influence of the interfacial roughnesses. A finite element (FE) model of crack propagation in the model system was developed and compared to experimental series. The crack direction in the FE model is calculated by using short trial cracks in different directions. The direction of the crack in the coating system is defined as the crack direction with the maximum energy release rate. It was found that microcracks form primarily near the roughness valleys and are more likely in the fast-creeping samples inside the TBC despite of the low creep strength. The higher probability of microcracks in these samples is because of the high energy release rate of initial delaminations. The highest life-time was measured in fast-creeping Fecralloy samples in combination with fast-creeping TGOs. The decelerated delamination growth rate in this sample is because of the interaction of delaminations with segmentation cracks. Therefore, an initial and fast-creeping TGO has the high potential to extend the life-time of the model systems. Hence, applying an initial and fast-creeping TGO on a fast-creeping BC is a possible optimization strategy for real thermal barrier coating systems

Determining Young’s modulus of coatings in vibrating reed experiments using irregularly shaped specimens

Knowing Young’s modulus of coatings is important to understand the stress evolution and failure of coating systems. However, such values are often not available for modern coatings and/or cannot be determined by existing methods for attaining a global stiffness. In this paper, a new method is described and validated that can be used even for brittle, highly porous, and irregularly shaped coatings, for example sputtered or thermally sprayed thermal barrier coatings: In vibrating reed experiments, the resonance frequencies of the specimens are determined. The specimen geometry is measured by a computer tomograph, and a finite element simulation based on that measured geometry is carried out to determine Young’s modulus. To validate this method, a monocrystalline irregularly shaped silicon plate with known Young’s modulus was measured. The method is tested on different metallic thermal spray coatings, for which other mechanical test methods for Young’s modulus were also applicable for comparison. Lastly, a very porous, gas flow sputtered zirconia thermal barrier coating was analyzed where other methods were not suitable

Another Look at Neural Multigrid

We present a new multigrid method called neural multigrid which is based on joining multigrid ideas with concepts from neural nets. The main idea is to use the Greenbaum criterion as a cost functional for the neural net. The algorithm is able to learn efficient interpolation operators in the case of the ordered Laplace equation with only a very small critical slowing down and with a surprisingly small amount of work comparable to that of a Conjugate Gradient solver. In the case of the two-dimensional Laplace equation with SU(2) gauge fields at beta=0 the learning exhibits critical slowing down with an exponent of about z = 0.4. The algorithm is able to find quite good interpolation operators in this case as well. Thereby it is proven that a practical true multigrid algorithm exists even for a gauge theory. An improved algorithm using dynamical blocks that will hopefully overcome the critical slowing down completely is sketched.Comment: 13 pages, 3 ps figures, uses IJMPC styl

Braunschweiger Schriften des Maschinenbaus, Band 8

Dieses Buch erläutert die verschiedenen numerischen Verfahren, die in der Materialwissenschaft zum Einsatz kommen. Die Auswahl der Verfahren erhebt dabei keinen Anspruch auf Vollständigkeit, bietet jedoch einen Querschnitt über die wichtigsten und am häufigsten verwendeten Techniken. Wegen ihrer großen Bedeutung in den Ingenieurwissenschaften wird die Finite-Element-Methode besonders umfassend behandelt. Zu den weiteren behandelten Methoden zählen zelluläre Automaten, Monte-Carlo-Methoden, Versetzungssimulationen, Molekulardynamik-Methoden, die Berechnung von Phasendiagrammen, neuronale Netze und Grundlagen der Bildbearbeitung und -analyse. Das Buch richtet sich an Studenten der Ingenieur- und Naturwissenschaften, die über Grundwissen in der Materialwissenschaft verfügen. Vorkenntnisse im Bereich der Numerik sind nicht erforderlich, die benötigten Methoden werden während der Diskussion der einzelnen Verfahren eingeführt. Die verwendeten Techniken werden an Beispielen erläutert und veranschaulicht