1. Collaboration with clinical medicine
We are collaborating with physicians to find solutions to medical problems based on mathematical science. This research is supported as a CREST project by the Japan Science and Technology Agency (JST). Our targets include the following.
- Blood flow in a human body is strongly related to various life-threatening diseases such as aortic aneurysms and aortic dissections. From a fluid dynamical perspective, instantaneous streamlines, secondary flows obtained by subtracting the axial flow from the total flow, helicity (which is an inner product of velocity and vorticity), the second largest eigenvalue of the rate of strain tensor, etc. provide useful information.
- Cerebrospinal fluid (CSF) flow around our brain and spine contributes to the health and safety of our central nervous system.
- Medical image diagnosis and estimation of clinically important parameters assisted by statistical techniques, which aims at extraction of essential algorithms from accumulated experience of medical doctors.
This project will contribute to high-performance clinical diagnoses through construction of decision-making tools including mathematical modeling, simulation technology, statistical analysis, image processing, and inverse analysis.
2. Mathematical approach to environmental problems
We are collaborating with environmental scientists, agricultural scientists, limnologists and many other scientists and engineers working to resolve environmental issues. Here also, our main tools are mathematical modeling and numerical simulations, which can be powerful tools for ascertaining behaviors related to environmental problems. Our targets include the following.
- Air flows from local-scales to meso-scales, are related to weather forecasts, prediction of natural disasters, and boundary conditions for water currents.
- Flows around aquatic plants and fluid-structure interactions between them play important roles in coastal ecosystems.
- Water currents in ponds, lakes and inland-seas are driven by winds, inflowing rivers, tides, and other forces, which affect water circulation, movements of nutrients or toxic materials, and daily life.
- Vertical circulation induced in a pond by water purification equipment.
- Tidal flows in inland seas are important for prediction and countermeasures related to natural disasters.
- Underground water flow and behavior of toxic materials in them.
In addition to domestic collaboration, we are collaborating with researchers from the Finnish Environment Institute supported by the Japan Society for the Promotion of Science (JSPS).
3. Numerical simulation methods
Widely various numerical methods are used, such as finite element, finite difference, and finite volume methods. We are working out the most appropriate means of simulations depending on the targeted problems. The whole procedure from mathematical modeling to programming by C, Python, FORTRAN, etc. via discretization processes, should be connected seamlessly. Our targets, in addition to #1 and #2 above, include flows around candles
, flows around rotating rugby footballs
, and sound fields around obstacles
4. Scientific visualization methods
Because the results of computational simulations are rows of numerical values, visualization techniques have been important tools for use with computational simulations. Scientists have been presenting visualizations of their results obtained using 1D graphs, 2D contour lines, 3D contour surfaces and volume rendering techniques
, 2D/3D vector arrows, 2D/3D passive particle paths
, etc. including photo-realistic rendering techniques and ray-tracing algorithms
mainly developed in computer graphics (CG) fields. In recent years, a new generation of scientific visualization techniques called "virtual reality" has been proposed. Several such technologies exist such as 3D/4D (3D + time axis) stereoscopic vision on screens with and without special eyewear. We are applying several visualization techniques in our daily research work and are finding ways of producing comprehensible renditions of our numerical results.