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Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 1 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
1. Mathematical Modelling
• describe a given problem with some mathematical formalism inorder to
– get a formal and precise description
– see fundamental properties due to the abstraction
– allow a systematic treatment and, thus, solution
• (mathematical) model: formal description (and usually simplifi-cation) of (some) reality
1.1. Example: Biofilms in Wastewater Treatment
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 2 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
Effects/phenomena Modelled Effects
In the fluid:
• wastewater flow
• pollutant transport
• chemical reactions
Microbes/Bacteria:
• metabolic activity
• competition
Interaction:
• changing fluid properties?
• formation of heterogeneousgeometries (sedimentation)
For example:
• fluid dynamics
• convective-diffusive-reactivepollutant transport
• biofilm growth (e.g. by cellu-lar automaton)
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 3 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
• bigger or smaller evidence:
– exact natural science and engineering: long tradition (basicconservation laws of continuum mechanics, e.g.)
* Navier-Stokes equations for fluid flow
* convection-diffusion equation for pollutant transport withina fluid
* diffusion-reaction equation for pollutant concentration withinthe biomass
– economics, game theory, climate modelling, modelling ofbiological phenomena: many open questions
* (Stochastic?) rules for biofilm growth?
* Modell for cell-cell communication?
* Modell for dithering microbes?
two parts of modelling: derivation and analysis
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 4 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
2. Derivation of a Model
2.1. Level of Detail
What do you want to model?
– the input-output relation of a biofilm reactor or the detailedphysical, chemical and biological processes within the biofilm
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 5 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
2.2. Relevant Quantities
Which are the important quantities for the task of your simula-tion?
– Biofilms:
* flow velocities,
* nutrient concentrations,
* temperature,
* porousity of the biofilm,
* exact shape of the biofilm surface,
* amount or spatial distribution of species and other com-ponents,
* mechanical properties of the biofilm (elasticity, viscos-ity),
* dithering of single microbes,
* ’communication’ between microbes?
– How important are they?
* Think of consequences of a neglection!
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 6 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
2.3. Relations
What are the relations and interactions between the relevantquantities?
– qualitative(x = c · y) andquantitative(c =?) aspects
• How can these be (mathematically) described?
– evolution of averaged pollutant concentration ⇒ ordinarydifferential equation
– fluid flow ⇒ partial differential equations
– initial or boundary conditions ⇒ algebraic equations
– non-negativity of pollutant concentration⇒ algebraic inequal-ity
– state transitions of microbes ⇒ automata
– order of several steps ⇒ graphs
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 7 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
3. Analysis of Models
3.1. Resulting Task
• Answer the question if there is a solution (Hamiltonian way in agraph)!
• Find a/the solution (flow field, pollutant concentration)!
• Find a/the best solution (optimization of a biofilm plant)!
3.2. Analysis of Solutions
• Does a solution exist?
• Is it unique?
• How does it depend on input data (discontinuously, continu-ously)?
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 8 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
3.3. Numerical Solvability
• condition numbers?
• complexity of finding a solution?
• known algorithms for approximating the solution?
3.4. Validation
• Is the model derived so far correct?
• validation with the help of experiments!
Mathematical Modelling – General Remarks
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 9 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
4. Classification of Mathematical Models
4.1. Discrete Models vs. Continous Models
• discrete modelsuse a discrete/combinatoric description (integernumbers, graphs,. . . )
• continuous modelsuse real quantities (real numbers, physicalquantities, differential equations,. . . )
∂~u
∂t+ (~u · ∇) ~u =
1
Re∆~u−∇p + ~g , (1)
0 = ∇T~u (2)
• primarily, but not necessarily: discrete models for discrete phe-nomena, continuous models for continuous phenomena (counterexamples: lattice-gas-automata for fluid flow, continuum me-chanics for traffic flow)
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 10 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
4.2. Deterministic Models vs. Stochastic Models
• deterministic models:
– input determines unique output
– reproducable results/simulations
• stochastic models:
– include random influences;
– simulations may produce different results for the same input
– usually averaged results of interest
• no general relation between phenomena and models:
– e.g. model diffusion as Brownian motion or as continuouseffect
– modelling of complex/unpredictable effects (weather/climatemodeling)
– modelling of varying input (car or network traffic)
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 11 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
Classification of Mathematical Models
• Discrete Models – Scheduling
• Discrete Models – Scheduling 2
• Discrete Models – Advanced Scheduling
• Discrete Models – Decision and Election Models
• Discrete Models – Modelling Election Procedures
• Discrete Models – Modelling Election Procedures 2
• Continuous Models – Population Models
• Continuous Models – Population Models 2
• Continuous Models – Population Models 3
• Continuous Models – Population Models 4
• Continuous Models – Heat Conduction
• Continuous Models – Heat Conduction 2
• Continuous Models – Concluding Remarks
• Stochastic Models (german)
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 12 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
4.3. Hierarchy and Multiscale Property of Models
• Choose scale or level of observation:
– Which resolution is necessary (w.r.t. the model’s accuracy)?
* turbulence – which vortices can be neglected?
– Which resolution can be tackled numerically?
– How many dimensions have to be or can be handled?
* biofilm simulation: · 1D: neglect biofilm hetrogeneity
· 3D: full resolution of all spatial effects
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 13 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
• Compute different effects on different levels
– spatial resolution only necessary for some model compo-nents
– averaging / homogenization of fine levels effects → quanti-tative influence on large scale model
– reduce dimensions to increase resolution? (exploit symme-tries)
– example turbulence
* significant transport of energy between different scales
* direct simulation – Large Eddy Simulation – averagingmodels
– example biofilm
* fine grain processes at the surface of substratum spherescrucial for the performance of the reactor
* coupling of fine and large scale via boundary conditionsat these surfaces
Multiscale and Hierarchical Models
• Multiscale Models
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 14 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
4.4. Averaging and Homogenization
• often: coarse-grain phenomena are of interest, but fine-grainphenomena must not be neglected
• try to do some averaging:
– in time: turbulence, molecular dynamics
– in space: flow and transport through porous media (a cata-lyst or soil)
• formal concept: homogenization
– representative elementary volume
– scaled reproduction, translation, periodic continuation
– limit process of scaling factor
– new quantities (effective parameters: porosity, permeabil-ity)
– new equations (porous media: instead of transport equa-tions now Darcy-Forchheimer equation)
Averaging and HomogenizationAveraging and Homogenization 2
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 15 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
5. 1st Example: Modelling of fluid dynamics
Discrete Models
• Lagrangian approach: fluid as set of interacting particles(could f.e. lead to system of ODE)
• Eulerian approach: particles moving within a given mesh(→ Lattice-Boltzmann automata)
Continous Models
• Navier-Stokes equations (system of PDEs);density, velocity, and pressure as functions
• discretization (Lagrangian and Eulerian approaches);leads to discrete models again
Stochastic vs. Deterministic Modeling:
• model diffusion as Brownian motion(not necessarily on the correct scale)
• allow random effects f.e. in Lattice-Boltzmann automata
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 16 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
How many dimensions?
• full 3D resolution necessary or wanted?
• exploit symmetries (rotational, axial, . . . ) to reduce dimensions
• average over one dimension (no vertical resolution)
• stationary or time dependent simulation?
Choose resolution
• desired accuracy vs.
• requirements from numerics
Multiscale and Hierarchical Modelling:
• resolve the geometry (averaging over fine structures)
• esp. in fluid flow: turbulence modelling
– significant transport of energy between different scales
– direct simulation (DNS) → Large Eddy Simulation (LES) →averaging models (RANS, k-ε, . . . )
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 17 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
6. 2nd Example: Modelling of biofilm growth
Discrete Models
• Lagrangian approach: microbes as moving particles within thefluid
• Eulerian approach, f.e. cellular automaton: (rectangular) cellsfilled with microbes of a common state (alive, dead, hungry, . . . )
Continous Models
• concentration of microbes and pollutants
• density/porosity of sediments
Stochastic vs. Deterministic Modeling:
• random walk models for microbes/bacteria
• allow random effects to simulate external influences (death of amicrobes)
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 18 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
How many dimensions?
• same resolution as for fluid dynamics?
• global concentration of microbes (same concentration every-where)
Choose resolution
• model any single microbe??
• local groups of microbes
• for cellular automaton: just give states for microbes, or alsoconcentrations?
Multiscale and Hierarchical Modelling:
• resolve geometry (averaging over fine structures)
In general: Population Modelling
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 19 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
7. 3rd Example: Modelling Interactions
Static or pseudo-static approach:
• Compute stationary flow field, and use result to simulate biofilmgrowth, or vice versa
• Compute stationary flow field, simulate biofilm growth during asmall time step, compute changes on flow field, etc.
Coupled equations:
• explicite time-stepping:
– compute changes over small(!) time steps
– interaction via intermediate results
• implicite time-stepping(solve system of equations to reach consistent state after eachtime step)
• fully coupled simulation:e.g.: extend system of differential equations to include all presenteffects
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 20 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
8. Setting Up Simulation “Experiments”
• simulation has to be “embedded”
• input of data, visualization, feedback to modelling/design
Example: biofilm modelling
Mathematical Modelling
Derivation of a Model
Analysis of Models
Classification of . . .
1st Example: . . .
2nd Example: . . .
3rd Example: . . .
Setting Up Simulation . . .
What to do with models?
Page 21 of 21
Introduction to Scientific Computing
2. Mathematical ModellsMiriam Mehl
9. What to do with models?
• the analytical approach:
– prove existence and uniqueness formally
– construct or find solution(s) formally/directly/analytically
– desirable, but almost never possible
• the heuristic approach:
– trial and error, following some (hopefully smart) strategy
– useful in discrete problems (travelling salesman etc.)
• the direct numerical approach:
– follow some numerical algorithm and end up with the exactsolution (Simplex algorithm for linear programming)
• the approximative numerical approach:
– approximate/discretize the model equations and end up withsome approximate solution