Dr. Martin Lenz

Dr. Martin Lenz
Address: Institut für Numerische Simulation
Endenicher Allee 60
53115 Bonn
Office: EA60 2.037
Phone: +49 228 733416

Research Projects

DFG priority program 1239: Change of microstructure and shape of solid materials by external magnetic fields
Project A6: Mathematical modeling and simulation of microstructured magnetic-shape-memory materials

The macroscopic behavior of magnetic shape-memory (MSM) materials is largely determined by the formation of fine-scale structures, both in the elastic and in the magnetic degrees of freedom. Most practically used samples exhibit additional inhomogeneities on the mesoscale, such as for example the grain structure in polycrystals. The interaction of the two types of microstructure is up to now only poorly understood. Focusing on two occurrences of high current experimental relevance we intend to investigate the role of microstructure for the macroscopic material behavior, and to furnish criteria to improve material production. Building upon the static continuum model developed in the first phase, which permits to resolve the magnetic and elastic structure at scales much smaller than the domain size, we plan: 1) to consider the experimentally relevant case of grains larger than the domain size, where the magnetoelastic behavior of a grain is determined by averaging over the domains; 2) to study dynamics and in particular hysteresis by formulating an evolutionary problem which resolves the motion of individual domain boundaries; and 3) to study the dynamics for grains larger than the domain size, with an averaged version of the evolution model. For each issue we shall address the development of a model, the numerical implementation of the obtained models, and the application to the study of MSM-polymer composites and MSM polycrystals.

DFG SFB 1060: The Mathematics of Emergent Effects
Project C06: Numerical optimization of shape microstructures

This project deals with the two-scale optimization of elastic materials. It is well known that microstructures form when minimizing compliance or tracking type cost functionals, unless a penalty on the area of material interfaces is used. The optimal microstructures are well-understood and can be represented by nested laminates. The laminate construction is an analytically elegant tool but can hardly be reproduced in mechanical devices, nor is it observed in optimization problems posed in nature. Thus, the question arises how close one can get to the optimal design with constructible microstructures. To this end, different approaches will be investigated and compared, namely: microscopic rod models with varying rod thickness, microscopic geometries described by a finite set of parameters, and non-constrained interfaces on the micro-scale with a microscopic interface regularization. To measure the closeness both to the achievable optimal design within the considered class and to a globally optimal laminate design, an a posteriori error analysis will be developed. Here, concepts for a residual error estimation based on the Lagrangian formulation of the optimization problem will be picked up to derive a posteriori estimates for the macroscopic error in the parameters describing microscopic geometries. In addition, the error caused by the chosen microscopic geometric model will be quantified via a posteriori error analysis. These resulting error estimates will be used to implement adaptive algorithms to steer the necessary and sufficient refinement of the macroscopic grid on which the parameters for the microscopic geometries are given and on which the microscopic geometric model is selected. Furthermore, a phase field model will be developed to describe non-constrained material interfaces on the micro-scale, where a diffuse interface energy regularizes the microscopically optimal material design. For this model truly two-scale a posteriori error estimates for the resulting shape optimization problem will be developed. This error analysis will enable mesh adaptivity on both the macro-scale, where the microscopic elastic energy density is evaluated for a given macroscopic elastic displacement, and the micro-scale, where the microscopic energy in dependence of the local microscopic interface geometry is actually computed. Concerning the underlying material design the focus will be on thick elastic domains in 2D or 3D filled with a composite of two different elastic materials or with an elastic and a void phase. As an alternative, rod type models on the micro-scale will be considered. With respect to the physical model, we will mainly deal with linearized elasticity and incorporate nonlinear material laws in later stages of the project. Finally, this project aims at carrying over the two-scale analysis of elastic bulk material to thin elastic shells, where the (in general nonlinear) stored elastic energy of a shell depends on the relative shape operator. Optimization will be performed with respect to the geometry and the thickness of the shell and will later be extended to an optimization of the shell microstructure.

General Research Interests

Materials Science

Boundary Element Methods for Sharp Interface Models of Ostwald Ripening

Project Multiple Scales in Phase Separating Systems with Elastic Misfit of Harald Garcke (Regensburg), Barbara Niethammer (Oxford) and Martin Rumpf within DFG Priority Program 1095 Analysis, Modeling and Simulation of Multiscale Problems.

Phase separation of a binary alloy occurs in a homogeneous mixture of two metals that is instable due to the presence of two energetically favoured phases. In an initial cooling step called spinodal decomposition, many nuclei of one phase form within a background matrix of the other phase. In a second stage, the nuclei undergo competitive growth in order to reduce their surface energy. The resulting coarsening process is also known as Ostwald ripening or aging. If the difference in size of the atoms is not too large, they can fit on a common lattice, but at the cost of a distortion of the lattice. The elastic energy arising from this distortion is proportional to the volume of the particles and thus becomes increasingly important in later process stages.

There are basically two types of models describing the behaviour of these coarsening processes. Either one studies the evolution of a sharp particle interface (Mullins—Sekerka model), or phase field models are investigated to describe a microscopic transition region at the interface (Cahn—Hilliard model). I study sharp interface models numerically and aim for an understanding of the different inherent scales. Furthermore, the complex elastic interactions have to be captured and their effect on the aging process has to be understood. Therefore, boundary element methods are applied. In particular, attributes that are significant for the material properties of the resulting alloy have to be identified, this includes the anisotropic particle shapes, alignment, attraction and repulsion of particles as well as the macroscopic coarsening rate.

For an effective macroscopic study of large-scale systems, models reduced to the essential degrees of freedom of a single particle are introduced. Hence, very large ensembles of particles can be simulated and statistically analysed. E.g. approximately rectangular shapes occur in the presence of a cubically anisotropic elasticity. Detailed comparison of the full and reduced models will give a quantified justification of the reduced models. Furthermore, multiscale acceleration techniques that exploit certain screening properties of the inter-particle interaction are applied.

Modeling and Simulation of Microstructured Magnetic Shape-Memory Materials

Project Modeling microstructured magnetic-shape-memory devices, with Sergio Conti (Duisburg-Essen) and Martin Rumpf, within DFG Priority Program 1239 Change of Microstructure and Shape of Solid Materials by External Magnetic Fields.

Composites of small magnetic-shape-memory particles embedded in a polymer matrix have been proposed as an energy damping mechanism and as actuators. Compared to a single crystal bulk material, the production is simpler and more flexible, as both type of the polymer and geometry of the microstructure can be tuned. Compared to polycrystals, in composites the soft polymer matrix permits the active grains to deform to some extent independently; in particular the rigidity of grain boundaries arising from incompatible orientation is reduced. I study the magnetic-field-induced deformation of composites, on the basis of a continuous model incorporating elasticity and micromagnetism, in a reduced two-dimensional, plane strain setting. Based on homogenization theory, I investigate the macroscopic behaviour by studying an affine-periodic cell problem. An energy descent algorithm is developed, whose main ingredients are a boundary element method for the computation of the elastic and magnetic field energies; and a combinatorial component reflecting the phase transition in the individual particles, which are assumed to be of single-domain type.

Fluid Dynamics

Finite Volume Methods for Thin Film Flow

Lubrication models allow the effective approximation of fluid dynamics in thin liquid layers on surfaces where phenonema like surface tension, evaporation, condensation and van der Waals forces are treated as driving forces. They describe the motion of the liquid—gas—interface and the contact lines between dry and wetted regions via nonlinear degenerate partial differential equations.

In a number of important applications, the evolution of a thin liquid film is mainly driven by surface tension effects (capillarity and Marangoni convection). In this case, finite volume methods are well suited, especially if the convection effects are dominant.

Capillarity is the consequence of the surface tension's absolute value, which tends to form drops minimizing surface tension, while the Marangoni effect due to gradients in surface tension induces a flow from regions of lower to higher surface tension. The Marangoni effect can be especially pronounced in the presence of surfactants (i.e. surface active agents), a substance forming a monomolecular layer on the film surface that reduces its surface tension. Such thin films occur in a number of important application fields, ranging from biology (e.g. the liquid lining of the lung's alveoli) to engineering (aircraft de-icing films).

The simulation of thin film flow driven by a surfactant has been the subject of my Diplom thesis. In this case, the description of the film height and the surfactant concentration are closely coupled. Operator splitting schemes are considered to reflect the multiple qualitatively different effects responsible for the evolution. This allows to properly treat the nonlinear Marangoni-type transport, especially if it is the dominating effect, while at the same time exhibiting the typical drop shapes due to the capillarity effect.


Efficient Visualisation Methods for Large Data Sets in GRAPE

The software package GRAPE has been developed at the Collaborative Research Center 256 at the University of Bonn and at the Institute for Applied Mathematics at the University of Freiburg. My main interests in scientific visualisation using GRAPE are adaptively hierachical postprocessing and visualization methods for large data sets accessed via a procedural interface.

A number of these methods are included in the postprocessing and visualization tool that has been developed in cooperation with the Gesellschaft für Anlagen- und Reaktorsicherheit in three projects funded by the German Federal Ministry of Education and Research, Entwicklung eines schnellen Programms zur Modellierung von Grundwasserströmungen mit variabler Dichte (d3f), Entwicklung eines Programmes zur dreidimensionalen Modellierung des Schadstofftransportes (r3t) and Weiterentwicklung der Rechenprogramme d3f und r3t (E-DuR).

Short Curriculum Vitae

Jan 2008 —Akademischer Rat z. A. at the Institute for Numerical Simulation
Dec 2002 — Dec 2007PhD Student in Mathematics at University of Duisburg—Essen and University of Bonn
PhD Thesis on Modellierung und Simulation des effektiven Verhaltens von Grenzflächen in Metalllegierungen
Supervisors: Martin Rumpf and Sergio Conti
partially supported by DFG via priority programs 1095 and 1239
Aug 1998 — Mar 2002Scholarship by Cusanuswerks — Bischöfliche Studienförderung
Oct 1996 — Nov 2002Student of Mathematics and Computer Science at University of Bonn
Diplom in Mathematics at Institute for Applied Mathematics
on Finite-Volumen-Methoden für degenerierte parabolische Systeme — Ausbreitung eines Surfactant auf einem dünnen Flüssigkeitsfilm
Supervisor: Martin Rumpf



[1] S. Conti, B. Geihe, M. Lenz, and M. Rumpf. A posteriori modeling error estimates in the optimization of two-scale elastic composite materials. ESAIM: Mathematical Modelling and Numerical Analysis, 2017. to appear.
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[2] S. Conti, M. Lenz, and M. Rumpf. Hysteresis in magnetic shape memory composites: Modeling and simulation. J. Mech. Phys. Solids, 89:272-286, 2016.
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[3] B. Geihe, M. Lenz, M. Rumpf, and R. Schultz. Risk averse elastic shape optimization with parametrized fine scale geometry. Mathematical Programming, 141(1-2):383-403, 2013.
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[4] S. Conti, M. Lenz, and M. Rumpf. Modeling and simulation of large microstructured particles in magnetic-shape-memory. Advanced Engineering Materials, 14(8):582-588, 2012.
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[5] M. Lenz, S. F. Nemadjieu, and M. Rumpf. A convergent finite volume scheme for diffusion on evolving surfaces. SIAM Journal on Numerical Analysis, 49(1):15-37, 2011.
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[6] S. Conti, M. Lenz, and M. Rumpf. Macroscopic behaviour of magnetic shape-memory polycrystals and polymer composites. In 7th European Symposium on Martensitic Transformations and Shape Memory Alloys, volume 481-482 of Materials Science and Engineering: A, pages 351-355, 2008.
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[7] M. Lenz, S. F. Nemadjieu, and M. Rumpf. Finite volume method on moving surfaces. In R. Eymard and J.-M. Hérald, editors, Finite Volumes for Complex Applications V, pages 561-576. Wiley, 2008.
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[8] S. Conti, M. Lenz, and M. Rumpf. Modeling and simulation of magnetic shape-memory polymer composites. Journal of Mechanics and Physics of Solids, 55:1462-1486, 2007.
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[9] M. Lenz. Modellierung und Simulation des effektiven Verhaltens von Grenzflächen in Metalllegierungen. Dissertation, University Bonn, 2007.
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[10] H. Garcke, M. Lenz, B. Niethammer, M. Rumpf, and U. Weikard. Multiple scales in phase separating systems with elastic misfit. In A. Mielke, editor, Analysis, Modeling and Simulation of Multiscale Problems. Springer, 2006. Final report of DFG priority program 1095.
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[11] J. Becker, G. Grün, M. Lenz, and M. Rumpf. Numerical methods for fourth order nonlinear degenerate diffusion problems. Applications of Mathematics, 47(6):517-543, 2002.
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[12] G. Grün, M. Lenz, and M. Rumpf. A finite volume scheme for surfactant driven thin film flow. In R. Herbin and D. Kröner, editors, Finite Volumes for Complex Applications III, pages 567-574. Hermes Penton Sciences, 2002.
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[13] M. Lenz. Finite Volumen Methoden für degenerierte parabolische Systeme - Ausbreitung eines Surfactant auf einem dünnen Flüssigkeitsfilm. Diploma thesis, University Bonn, 2002.
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[14] T. Geßner, B. Haasdonk, R. Kende, M. Lenz, R. Neubauer, M. Metscher, M. Ohlberger, W. Rosenbaum, M. Rumpf, R. Schwörer, M. Spielberg, and U. Weikard. A procedural interface to hierarchical grids. Technical report, SFB 256, University Bonn, 1999.
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