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Remember: The Simulation Pipeline
phenomenon, process etc.
mathematical model?
modelling
numerical algorithm?
numerical treatment
simulation code?
implementation
results to interpret?
visualization
ļæ½ļæ½ļæ½ļæ½ļæ½HHHHj embedding
statement tool
-
-
-
validation
Michael Bader | Scientific Computing II | Overview | Summer 2017 2
Topic #1: Solving Systems of Linear Equations
Focussing onā¢ large systems: 106ā109 unknownsā¢ sparse systems: typically only O(N) non-zeros in the system matrix
(N unknowns)ā¢ systems resulting from the discretization of PDEs
Topicsā¢ relaxation methods (as smoothers)ā¢ multigrid methodsā¢ Conjugate Gradient methodsā¢ preconditioning
Michael Bader | Scientific Computing II | Overview | Summer 2017 3
Recall: Finite Volume Model for Heat Equationā¢ object: a rectangular metal plateā¢ model as a collection of small connected rectangular cells
hx
hy
ā¢ compute the temperature distribution on this plate!
Michael Bader | Scientific Computing II | Overview | Summer 2017 4
A Finite Volume Model (2)
ā¢ model assumption: temperatures in equilibrium in every grid cellā¢ heat flow across a given edge is proportional to
ā¢ temperature difference (T1 ā T0) between the adjacent cellsā¢ length h of the edge
ā¢ e.g.: heat flow across the left edge:
q(left)i,j = kx
(Ti,j ā Tiā1,j
)hy
note: heat flow out of the cell (and kx > 0)ā¢ heat flow across all edges determines change of heat energy:
qij = kx(Tij ā Tiā1,j
)hy + kx
(Tij ā Ti+1,j
)hy
+ ky(Tij ā Ti,jā1
)hx + ky
(Tij ā Ti,j+1
)hx
Michael Bader | Scientific Computing II | Overview | Summer 2017 5
A Steady-State Model. . . and a large system of linear equations
ā¢ heat sources: consider additional source term Fi,j due toā¢ external heatingā¢ radiation
ā¢ Fi,j = fi,jhxhy (fi,j heat flow per area)ā¢ equilibrium with source term requires qi,j + Fi,j = 0:
fi,jhxhy = ākxhy(2Ti,j ā Tiā1,j ā Ti+1,j
)āky hx
(2Ti,j ā Ti,jā1 ā Ti,j+1
)ā¢ leads to large system of linear equationsā¢ 1/h2 unknowns, sparse system matrix (only 5 entries per row)
ā will be our model problem!
Michael Bader | Scientific Computing II | Overview | Summer 2017 6
Multigrid: HHG for Mantle Convection(Rude et al., 2013; project: TERRA NEO)
Michael Bader | Scientific Computing II | Overview | Summer 2017 7
Multigrid: HHG for Mantle Convection (2)Mantle Convection on PetaScale Supercomputers:
ā¢ mantle convection modeled via Stokes equation (ācreeping flowā)ā¢ linear Finite Element method on an hierarchically structured tetrahedral
meshā¢ requires solution of global pressure equation in each time step
Weak Scaling of HHG Multigrid Solver on JuQueen:ā¢ geometric multigrid for Stokes flow via pressure-correctionā¢ pressure residual reduced by 10ā3 (A) or 10ā8 (B)
Nodes Threads Grid points Resolution Time: (A) (B)1 30 2.1 Ā· 1007 32 km 30 s 89 s4 240 1.6 Ā· 1008 16 km 38 s 114 s
30 1 920 1.3 Ā· 1009 8 km 40 s 121 s240 15 360 1.1 Ā· 1010 4 km 44 s 133 s
1 920 122 880 8.5 Ā· 1010 2 km 48 s 153 s15 360 983 040 6.9 Ā· 1011 1 km 54 s 170 s
Michael Bader | Scientific Computing II | Overview | Summer 2017 8
Topic #2: Molecular Dynamics
Discuss large part of the simulation pipeline:ā¢ modelling: potentials, forces, systems of ODEā¢ numerics: suitable numerical methods for the ODEsā¢ implementation: short-range vs. longe-range forcesā¢ visualisation? (well, actually not the entire pipeline . . . )
Focussing onā¢ large systems: 106ā109 particlesā¢ short-range vs. long-range forcesā¢ N-body methods, parallelization
Michael Bader | Scientific Computing II | Overview | Summer 2017 9
N-Body Methods: Millennium-XXL Project
(Springel, Angulo, et al., 2010)
ā¢ N-body simulation with N = 3 Ā· 1011 āparticlesāā¢ compute gravitational forces and effects
(every āparticleā correspond to ā¼ 109 suns)ā¢ simulation of the generation of galaxy clusters
plausibility of the ācold dark matterā modelMichael Bader | Scientific Computing II | Overview | Summer 2017 10
N-Body Methods: Particulate Flow Simulation
(Rahimian, . . . , Biros, 2010)
ā¢ direct simulation of blood flowā¢ particulate flow simulation (coupled problem)ā¢ Stokes flow for blood plasmaā¢ red blood cells as immersed, deformable particles
Michael Bader | Scientific Computing II | Overview | Summer 2017 11
Exams, ECTS, Modules
ECTS, Modulesā¢ 5 ECTS (2+2 lectures/tutorials per week)ā¢ CSE: compulsory courseā¢ Biomed. Computing/Computer Science:
elective/Master catalogueā¢ others?
Tutorials:ā¢ tutor: Carsten Uphoffā¢ time and day: Tue, 10-12, MI 02.07.023
Exam:ā¢ written exam at end of semesterā¢ based on exercises presented in the tutorials
Michael Bader | Scientific Computing II | Overview | Summer 2017 13
Lecture Slides: Color Code for Headers
Black Headers:ā¢ for all slides with regular topics
Green Headers:ā¢ summarized details: will be explained in the lecture, but usually not as an
explicit slide; āgreenā slides will only appear in the handout versions
Red Headers:ā¢ advanced topics or outlook: will not be part of the exam topics
Michael Bader | Scientific Computing II | Overview | Summer 2017 14