Matematisk modelering og simulering

1. Matematisk modelering og simulering Hans Petter Langtangen Simula Research Laboratory Dept. of Informatics, Univ. of Oslo

2. Questions I will address • What is mathematical modeling and simulation, or computational science? • Why is it so important? • When/where is it useful? • What kind of competence and knowledge is needed?

3. The story of the success of simulation is a story about making mathematics much more useful

4. The role of mathematics cannot be underestimated • For the last 300 years, mathematics has been a key tool in the development of science and technology • Result: dramatically higher living standards • Mathematics will be dramatically more useful in the future • Why? Because of fast computers • This fact will accelerate science and technology

5. Why computers make mathematics more useful • We have equations (mathematical models) for how nature works • Nature = weather, climate, oil production, airplane manufacturing, space exploration, epidemics, wireless communication, … • The difficulty is to solve the equations • Before the computer: specialized methods with pen & paper for a few problems • After modern, fast computers: general and widely applicable solution methods

6. It is impossible to exaggerate the extent to which modern applied mathematics has been shaped and fuelled by the general availability of fast computers with large memories. Their impact on mathematics, both applied and pure, is comparable to the role of telescopes in astronomy and microscopes in biology... I am on the safest ground in surmissing that computing will play en even bigger role in the next century than today. P. D. Lax, SIAM Review 1989

7. All modern weather forecasts are based on extensive simulation

8. Weather forecast in Norway Horizontal resolution: 4 km 300 x 500 x 38 grid points Time step: 1 min Simulation period: 60 h Determines parameters in 20.5 billion points Roar Skålin, IT manager, met.no

9. The computer becomes a lab • We need to make a program to do mathematics on a computer • Complicated mathematical models and solution methods can be packed in user-friendly software and ”hidden” • With such software, the computer becomes a cheap laboratory for experimentation! • Using the computer as a lab is often called simulation • Here we look at simulation based on mathematical models

10. Exploration software

11. Now we do computational science • Wikipedia definition: • Computational science is the use of computers to perform research in other fields. It is the application of computer simulation and other forms of computation to problems in various scientific disciplines. • It is not to be confused with computer science which is the study of topics related to computers and information processing.

12. Related terms • Computational science and engineering • Mathematical modeling and simulation • Numerical modeling • Scientific computing (study, implement and apply algorithms) • Numerical mathematics (study properties of algorithms)

13. Computational science has become the third pillar of the scientific enterprise, a peer alongside theory and physical experiment. Computational science is now indispensable to the solution of complex problems in every sector . Advances in computing...make it possible to develop computational models...to address problems previously deemed intractable. PITAC Report , ”Ensuring America’s Competitiveness”, to the US president, 2005

14. The next 10 to 20 years will see computational science firmly embedded in the fabric of science – the most profound development in the scientific method in over three centuries. US Department of Energy, 2003

15. DATASET UNSTRUCTURED_GRID POINTS 201 float 2.77828 2.18262 -0.25 0.476 2.4 -0.85 0.85 2.4 -0.476 -0.476 2.4 -0.85 -0.85 2.4 -0.476 -0.85 2.4 0.476 -0.476 2.4 0.85 0.476 2.4 0.85 0.85 2.4 0.476 2.55 0.8625 0.66 CELLS 458 2290 4 41 29 65 80 4 53 41 65 82 4 35 34 47 71 Prediction & Control Results Refinement Processes Computations Mathematical Model The Simulation Pipeline

16. Simulation vs. real experiments • Simulation is cheap compared to physical experiments (lab or field) • Physical experiments may be dangerous, impossible or too expensive • Simulations give more detailed information and understanding • The best is to do both!

17. Past, present and future applications of simulation • Weapons, logistics, space exploration • Classical industry (structural, car, ship, airplane, oil & gas, chemical, consumer products, …) • Electronics, telecommunications • Advanced materials (incl. nanotechnology) • Construction of new molecules (chemestry on computer) • Environmental research, incl. climate predictions • Medicine: surgery, diagnostics • Geological evolution of the earth (e.g., oil reservoirs) • Evolution of planets, galaxies, universe • Biological processes and evolution • Sociological, psycological, economical processes

18. Warm winters and cold summers Each frame in this animation of the surface temperature of the Gulf Stream represents a seven day period.

19. Tsunamis in fjords Knut-Andreas Lie, SINTEF

20. The tsunami in the Indian Ocean, Dec 26, 2004 Jan Olav Langseth Dave George Randy LeVeque ”Mesh level 1” 111 km x 111 km ”Mesh level 3” 1.7 km x 1.7 km ”Mesh level 4” 25 m x 25 m

22. Simulation is a key tool in the aerospace industry

23. Crashing cars in the computer is cheaper than in reality

24. Unstructured grids ”High lift configuration” CRAY T3E – 1450 processors, 25 million gridcells University of Wyoming (1998)

25. Simulation is a key tool in studying the universe A comet, 1 km in diameter, entering Jupiter’s atmosphere at 134,000 miles per hour. (Red comet core of solid ice.)

26. Facts about the simulation • Turbulent structures • Gravity/temperature driven • 1 million CPU hours • 1000 processors • 100.000 GB of data Joe Werne, Colorado Research Associates DivisionNorthWest Research Associates, Inc.

28. Simulation is a key tool in the oil & gas industry

29. Oil-water flow in oil reservoirs Knut-Andreas Lie, SINTEF

30. New understanding of life processes Simulation is important in the exploration of life processes, ranging from studies of DNA to investigations of blood circulation and inner organs like the heart, brain and lungs.

31. DNA and Drug Design Better understanding of the structure of DNA may lead to new and improved drugs, like a vaccine for the flu!

32. 3D time-dependent Navier-Stokes simulations of the airflow in the lungs. Methods from aerospace and car industry are adapted to life sciences. What happens with smoke in your lungs?

33. Electrical activity in the heart: estimate infarctions by simulations

34. Blood flow simulation Martin Sandve Alnæs Tor Ingebrigtsen Jørgen Isaksen Kent-Andre Mardal Ola Skavhaug Univ. of Tromsø, Simula Research Lab.

38. Challenges in simulation: mathematics, algorithms, software • Multi-physics • Multi-scale • Multi-disciplinary • Multi-institutional code/teams • Obtaining real-life input data, e.g., complex geometries • Total system simulation (trees of complex simulation components)

39. Different types of mathematical models are used for different physical scales • Elementary particles: quantum mechanics Schrodinger equation, system of particles • Molecules: molecular dynamics System of particles; ordinary differential eqs. • Macro-scale: continuum mechanics Partial differential eqs.

40. Limitations of simulation • For some industrial processes (esp. structural analysis), mathematical models and simulation have high precision • In complex media (geology, medicine) lack of media details and complex physics may lead to low quantitative precision • Despite low precision, simulation may provide important insight into the physics • Simulation as a learning tool in combination with human experience and knowledge is often more useful than accurate prediction

41. Hardware vs algorithmic development 1970 - 2000 Updated version of chart appearing in “Grand Challenges: High performance computing and communications”, OSTP committee on physical, mathematical and Engineering Sciences, 1992.

42. Computing in Parallel Computing in Parallel Computing in Parallel Computing in Parallel • Simulation requires enormous computational power (speed, storage) • Processors get faster…(2x every 18 months) • …but a much larger gain in speed comes from coupling computers in parallel • Split a problem in subproblems and let many computers deal with subproblems in parallel • Requires computers to communicate • Humans think sequentially; constructing parallel algorithms is hard

43. Simulation software is more complex than most other software! • Very large program systems • Complicated mathematical models • Great algorithmic complexity • Difficult to test, complicated output • Extreme demands to • fast computations • memory usage • Fancy GUIs and colorful results…

44. Can anyone do simulation? • Simulation packages have become ”easy” to use and provide impressive colorful results • Result: ”anyone” can simulate! • However, without a thorough understanding of the mathematical model, it is easy to provide wrong input data, or ignore options • Judging the quality of the results is difficult

45. What can go wrong? • Lots of input data, usually with default values, but are the default values appropriate? • Picking the wrong mathematical model • Forgetting boundary conditions • Choosing an inadequate numerical solution method and/or associated parameters • Results consist of numerical artifacts and real physical features – what is what?

46. Wrong simulations may lead to very expensive disasters • A primary example is the Sleipner platform • Insufficient use of computations caused a structural failure and the platform sank • Cost: 700M $

47. What kind of competence do we need to do simulation? • Many can run simulation programs, but at least one in the team must understand the complexity of the model and pitfalls of the program’s simulation techniques • Education in simulation is immature and incomplete • This competence is emerging in new computational science & engineering university programs • To do high-quality simulations, one needs competence that take years to build systematically • This competence building requires long-term strategic plans in R&D institutions

48. Programming builds competence in an effective way • Internal software development is an effective and simple exercise to build competence • ”Programming is understanding” (K. Nygaard) • Even if a sophisticated external software package is to be used for production simulation, programming a simplified model is a specific way to gain insight into the model and relevant numerical techniques • Programming is expensive, but building competence is expensive, and delivering wrong simulation results is even more expensive…

49. Summary • Simulation (computer=lab) is a now key tool in science and technology • Every project should investigate the possibilities offered by simulation! • Better numerics and faster hardware will make simulation even more important • Simulation involves advanced mathematics, physics, +++ and requires high competence • The success of simulation relies on sucess in proper competence building • Programming = efficient competence building