1. mathematical modelling - in.tum.de filemathematical modelling derivation of a model analysis of...

Mathematical Modelling

Derivation of a Model

Analysis of Models

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Setting Up Simulation . . .

What to do with models?

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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

http://www5.in.tum.de/persons/mehl/



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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)




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• 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




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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




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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!




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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




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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)?




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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


http://wwwcse.informatik.tu-muenchen.de/vtc/SciComp/main/ISC2_1.html



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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)




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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)




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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 – Heat Conduction

• Continuous Models – Heat Conduction 2

• Continuous Models – Concluding Remarks

• Stochastic Models (german)



http://wwwcse.informatik.tu-muenchen.de/vtc/SciComp/main/ISC2_3_1.html

http://wwwcse.informatik.tu-muenchen.de/vtc/SciComp/main/ISC2_3_1a.html




http://wwwcse.informatik.tu-muenchen.de/vtc/SciComp/main/ISC2_3_2b.html




http://wwwcse.informatik.tu-muenchen.de/vtc/SciComp/main/ISC2_4_1c.html







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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




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• 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






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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






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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




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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, . . . )




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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)




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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




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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




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8. Setting Up Simulation “Experiments”

• simulation has to be “embedded”

• input of data, visualization, feedback to modelling/design

Example: biofilm modelling




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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


1. mathematical modelling - in.tum.de filemathematical modelling derivation of a model analysis of...

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