In many engineering applications, interest is restricted to a single quantity. For instance, in heat transfer applications, it is ultimately only the heat flux through a certain part of the boundary that is of interest. A crucial notion concerns the fact that the restricted interest to one particular goal functional can be exploited to reduce the complexity of numerical simulation methods. This notion is formalized by so-called goal-oriented a-posteriori error estimation and optimal adaptive-refinement methodologies.
By means of the solution of an appropriate dual problem, the contribution of local errors in the solution to the error in the goal functional can be established. Only the regions that have a pronounced influence on the error in the goal functional need to be refined in the numerical model. Such an approach can be used for both discretization adaptivity and model adaptivity. In the first form of adaptivity, the computational mesh (h) or order of approximation (p) is locally adapted to reduce the error. In the second form of adaptivity, the underlying model is locally adapted, e.g., by locally replacing a continuum model by a molecular model.
Airbags can cause severe injuries to a passenger if impact occurs before full deployment. To prevent such out-of-position situations, a precise understanding of the dynamics of the airbag is required. Numerical simulations can provide valuable information on the dynamical behavior of an airbag. However, the numerical simulation of airbag-deployment dynamics is a complicated endeavor, on account of the large range of length scales that the airbag traverses during the deployment process.
The initial stowed or folded configuration of the airbag can be characterized as a labyrinth of extremely thin folds. As the inflator is fired, the labyrinth is perfused by the inflator gas. The corresponding pressurization causes the folds to expand and, ultimately, the airbag to unfold into its bulbous final configuration. The perfusion and the corresponding expansion and unfolding constitute a multiscale fluid-structure-interaction problem of very high complexity, in which the microscale associated with the many tiny folds in the initial configuration is connected to the macroscale pertaining to the final configuration. Hence, the characteristic length scale of the airbag geometry changes by many orders.
One of the grand challenges of our times is to understand the mechanisms of cancer so that reliable treatments or preventative measures can be determined to relieve the impact this disease has on so many people. Cancer arises from one single normal cell transforming into a tumor cell. This tumor cell along with its progeny exhibit extensive proliferation (uncontrolled division) and form a growing mass of abnormal cells, i.e., a tumor arises.
The avascular growth of tumors can be described by continuum models that aim at predicting the evolution of large tumorous regions while ignoring the behavior of individual cells. These continuum models of tumor growth are derived from first principles through a continuum theory for multi-constituent media known as mixture theory. Most tumor-growth models are interface problems (free-boundary problems), and this can be modeled within mixture theory by including concentration gradients of various constituents into the Helmholtz free-energy functional. In this manner, one obtains diffuse-interface (or phase-field) models that introduce smooth transitional boundaries between the various constituents. The resulting equations are systems of the Cahn-Hilliard type, namely complex systems of nonlinear evolution fourth-order partial-differential equations, whose accurate numerical simulation poses enormous challenges to overcome.