Engineering Mathematics
|
Project leader: Michael Oberguggenberger
Institute for Basic Sciences in Civil Engineering - Unit of Engineering Mathematics
Mathematical and numerical methods in civil engineering rest on the pillars of partial differential equations (with finite element methods) and stochastics. In particular, stochastic and probabilistic methods have entered prominently in reliability analysis and sensitivity analysis, which are at the core of the design process of engineering structures. The term imprecise probability has been established to comprise interval computations, random sets, belief functions and interval valued probability, methods that gain increasing influence in uncertainty and reliability assessments as well as validation and verification in the engineering sciences. These methods require massive parallel computations to be efficient in large scale engineering problems. One of the central current research topics is the development of new and faster algorithms to be used in sensitivity analysis and computations of the stochastic response of a structure.
Among the many research topics suitable for a PhD dissertation, two projects are proposed that aim at merging stochastic, numerical and high performance computing methods.
- Monte Carlo methods in iterative solvers
- Random set models for parameter uncertainty in structural models driven by stochastic excitations
Students will benefit from a variety of international scientific and industrial collaborations as well as collaborations within the DK+, among them established joint research with A. Ostermann (numerical methods in sensitivity analysis), C. Adam (stochastic dynamics and random sets), G. Hofstetter (finite element methods) and T. Fahringer (parallelization).
Research Environment:
The Unit of Engineering Mathematics has a unique position as part of a civil engineering faculty. This implies a wide variety of research areas between theory, numerics and applications that are covered at the unit. Collaborations exist locally and internationally with academic, engineering and industrial partners.
Know-how of the research group:
Nonlinear partial differential equations;
stochastic analysis, probability theory and generalizations;
analysis of risk and reliability;
theory and numerics of Finite Elements;
Monte Carlo methods and stochastic simulations.
Engineering Mathematics:
|
Monte Carlo methods have been found highly appropriate for sensitivity and reliability analysis in engineering models, whose input-output function is given by a numerical code. A major issue is the computational cost. Sample sizes of at least 100 are required, e.g., for obtaining a ranking of the most influential input parameters, combined with numerical models in which a single input-output computation may take several hours or even days. Reduction of computational cost is mandatory. It is the aim of this project to develop new numerical methods that place the Monte Carlo simulation at a late stage in an iterative solver. This has been successfully tested in linear problems and in nonlinear problems of the load incremental type. In this dissertation, new iterative solvers should be developed for more general nonlinear large scale finite element models. On the one hand, this will require addressing the numerical codes at a deep level; on the other hand, these methods lend themselves to parallel and distributed computations, which should be incorporated at the core of the methods.
The figures below show a finite element model of a shell structure from aerospace engineering (from a joint project with the A. Ostermann and the company INTALES Engineering Solutions GmbH) as well as a picture of partial rank correlation coefficients measuring the influence of 17 input variables on an output variable describing buckling failure. The computations were based on Monte Carlo simulations.
The ideal candidate for this Ph.D. position should have prior knowledge in numerical analysis, finite element programs, and basic probability theory.
 |
 |
Project 1: Monte Carlo methods in iterative solvers
Engineering Mathematics:
|
Stochastic mechanics is based on stochastic (aleatoric) models for the external driving forces. On the other hand, the uncertainty of various input parameters may be of an epistemic nature, more appropriately modeled by random sets. The combination of the two sides has been successfully initiated for stochastic differential equations. In the field of stochastic finite elements, the combination of random sets and stochastic processes is a big open research topic. This will require new methodology as well as massive parallelization. Here the challenge comes from combining interval computations with large scale stochastic simulations. New methodology will come from a detailed investigation of how interval bounds can be propagated through stochastic models. Parallel computation will form a central ingredient. Important models to be developed are in the area of nonlinear soil dynamics. The stochastic input comes from earthquakes, while random sets are required to model parameter uncertainty.
The figures below shows the interval bounds of the relative displacement of a Tuned Mass Damper under stochastic excitation with uncertain coefficients (from joint research with C. Adam and B. Schmelzer). The model is described by a system of Itô stochastic differential equations; the excitation is a white noise process; parameter uncertainty is modeled by random sets. The numerical solution required both the simulation of the trajectories as well as covering the interval ranges for the coefficients.
The ideal candidate for this Ph.D. position should have prior knowledge in finite element methods and stochastic analysis.
