Port-based approach of complex distributed-parameter system models for analysis and simulation (PACDAS, STW-TWI.6012)

Criteria and goals

The research is bounded by a selection of criteria and goals. Many research fields cover one or more of these criteria. However, the combination of all of these criteria makes this project unique. Especially, modularity is not covered systematically by most approaches.

Modularity

In practice, systems consist of many parts. For a systematic modeling, these parts should be described and characterized independently, however, with the possibility to connect parts together, and to user input and output. "One man's output, is another man's input." The connection requires a systematic approach, which in this case is the power-based, or port-based, approach. Most modeling is based on the isolated model. In three dimensions, such models have many degrees of freedom irrelevant in operational conditions, input and output, and keeping track of the energy flow should characterize the dominant features of a component model.

Boundaries

In a mathematical model of a PDE boundaries are functions at the boundary, related to the interior through theorems like the divergence theorem and Stokes theorem. In practice the interface of a system with its surrounding through the boundary have to be matched from one system to the next. For example, already in the single-domain problem of the theory of light, a variety of reflectance-transmittance boundaries exists, which can depend on very fine details at the boundary.

Multidomain

With a solely mechanical, or electrical description of a system one will get not very far in industrial applications. Most systems combine two or more physical domains together. A proper description allows for all kinds of interactions between and among: mechanical, electrodynamical, thermal, diffusive, flow, radiation, control signal, chemical, and electrical domains.

Multiscale

Especially in the case of multi-domain problems, the spatial and temporal scales of the different dynamics can be very different. Proper modeling should make that apparent and handle these differences in simulations. An efficient approach will take care of the dynamics, or time-dependence, in the choice of scales and variables.

Nonlinearity

Nonlinearities appear in all kind of systems; linearizing might not always suffices. Furthermore, intrinsically linear conservation laws might apply, even in the case of nonlinear constitutive relations. These conservation laws are important to stabilize systems which are only weakly damped. In the case of model reduction, the reduction of an abstract and general nonlinearity to an effective nonlinear model requires special attention, to ensure, e.g., energy conservation and stability.

Model Reduction

A PDE of a dynamical system is in principle a infinite-dimensional model. For practical purposed of simulation and control, possibly optimal control, the model has to be reduced to a order which allows for fast implementation in simulation codes. Many reduction methods (POD, CVT) are based on snapshots of large model simulations for specific input. The dynamics in such models are based on some impulse response to generate a large "snapshot model space", hence will fail to describe models without predescribed input-output designation.

Optimality

Control of systems described by PDE, usually referred to as "optimal control" in the mathematical literature, falls, in practice, apart into two distinct sections: The design part and the control part. The control part is the low-dimensional time-dependence, regulated through some control and feedback loop between input and output, while the design part is the high-dimensional part, usually independent of particulars of the controller. The linkage between these two part should yield important and sophisticated design criteria, expressed in some optimality condition.

Implementation

Eventually, algorithms and software code should come out of this project which bridge the gap between design and simulation in several ways. Possibly it should link to existing FEM descriptions of systems, and produce reduced-order models which can be used in simulations. Furthermore, it should generate proper reduced-order models of standard systems, such as beam models and wave guides, without reference a larger model.