### Description

The temperature distribution of an object in a time-dependent process can be computed by solving the heat equation. Conventional simulation methods compute the changes in small time steps. This approach requires many sequential computations. In this way, it is not possible to take full advantage of the huge resources of recent supercomputers. Therefore such computations usually take a lot of time.

Space-time methods deal with the considered time interval as a whole. Therefore, a very large problem has to be solved instead of a sequence of smaller problems. This seems to be more difficult at first glance but offers enormous potential. Dealing with the full time interval as a whole makes possible an additional parallelization in time. Thus, the computation can be distributed to more processors and the simulation result can be computed much faster. In addition, it is possible to adapt the approximations of the temperature much better to the specific problem.

In boundary element methods it is sufficient to compute the considered data on the surface initially. Then the temperature can easily be evaluated in the interior. Although boundary element methods for the heat equation are space-time methods inherently, the related problem is usually still solved in small time steps by forward elimination. In doing so, the potential of the methods mentioned above is not exploited.

In this project, we develop fast methods for the solution of space-time boundary element methods to compute the temperature distribution of an object. These fast methods are specifically tailored to the space-time equations. For this purpose suitable preconditioning techniques and solution procedures are developed, as it is not straightforward to compute the current state without knowing the previous one. In addition, specifically designed representations of the wanted temperature are implemented to adopt the computation much better to the temperature distribution of the considered problem.

We put an extra emphasis on the parallel implementation of the methods on modern supercomputers. Modern processors enable the same computation on several sets of data at the same time by vectorization. In addition, they are equipped with several cores which can be used simultaneously. Supercomputers have a large number of computers which can be used in a cooperating way. We will utilize all three levels of parallelization in this project. This has to be taken into account in the development of the methods.

By the end of the project, it will be possible to compute the temperature distribution of an object much more accurately and faster than by currently used conventional methods.

Space-time methods deal with the considered time interval as a whole. Therefore, a very large problem has to be solved instead of a sequence of smaller problems. This seems to be more difficult at first glance but offers enormous potential. Dealing with the full time interval as a whole makes possible an additional parallelization in time. Thus, the computation can be distributed to more processors and the simulation result can be computed much faster. In addition, it is possible to adapt the approximations of the temperature much better to the specific problem.

In boundary element methods it is sufficient to compute the considered data on the surface initially. Then the temperature can easily be evaluated in the interior. Although boundary element methods for the heat equation are space-time methods inherently, the related problem is usually still solved in small time steps by forward elimination. In doing so, the potential of the methods mentioned above is not exploited.

In this project, we develop fast methods for the solution of space-time boundary element methods to compute the temperature distribution of an object. These fast methods are specifically tailored to the space-time equations. For this purpose suitable preconditioning techniques and solution procedures are developed, as it is not straightforward to compute the current state without knowing the previous one. In addition, specifically designed representations of the wanted temperature are implemented to adopt the computation much better to the temperature distribution of the considered problem.

We put an extra emphasis on the parallel implementation of the methods on modern supercomputers. Modern processors enable the same computation on several sets of data at the same time by vectorization. In addition, they are equipped with several cores which can be used simultaneously. Supercomputers have a large number of computers which can be used in a cooperating way. We will utilize all three levels of parallelization in this project. This has to be taken into account in the development of the methods.

By the end of the project, it will be possible to compute the temperature distribution of an object much more accurately and faster than by currently used conventional methods.

Status | Active |
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Effective start/end date | 1/03/19 → 28/02/22 |

### Keywords

- 101014 Numerical mathematics
- 102023 Supercomputing