1. Mathematical Modelling

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Charles Freeman
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1. describe a given problem with some mathematical formalism in order to get a formal and precise description see fundamental properties due to the abstraction allow a systematic

1 1. describe a given problem with some mathematical formalism in order 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 simplification) of (some) reality 1.1. Example: Biofilms in Wastewater Treatment Page 1 of 13 2.

2 Effects/phenomena Modelled Effects In the fluid: wastewater flow pollutant transport chemical reactions Microbes/Bacteria: metabolic activity competition Interaction: changing fluid properties? formation of heterogeneous geometries (sedimentation) For example: fluid dynamics convective-diffusive-reactive pollutant transport biofilm growth (e.g. by cellular automaton) Page 2 of 13 2.

2. Classification of Mathematical Models 2.1. Discrete Models vs. Continous Models discrete models use a discrete/combinatoric description (integer numbers, graphs,.

3 2. Classification of Mathematical Models 2.1. Discrete Models vs. Continous Models discrete models use a discrete/combinatoric description (integer numbers, graphs,... ) continuous models use real quantities (real numbers, physical quantities, differential equations,... ) u t + ( u ) u = 1 u p + g, Re (1) 0 = T u (2) primarily, but not necessarily: discrete models for discrete phenomena, continuous models for continuous phenomena Page 3 of 13 2.

4 2.2. 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 continuous effect Modelling of complex/unpredictable effects (weather/climate modeling) Modelling of different input (car or network traffic) Page 4 of 13 2.

5 2.3. Hierarchy and Multiscale Property of Models Choose scale or level of observation: Which resolution is necessary (w.r.t. the model s accuracy)? Which resolution can be tackled numerically? How many dimensions have to be or can be handled Different effects on different levels? spatial resolution only necessary for some model components Homogenization of fine level effects quantify influence on large scale model reduce dimensions to increase resolution? Page 5 of 13 2.

6 2.4. Averaging and Homogenization often: coarse-grain phenomena are of interest, but fine-grain phenomena must not be neglected try to do some averaging: in time: turbulence, molecular dynamics in space: flow and transport through porous media (a catalyst or soil) formal concept: homogenization representative elementary volume scaled reproduction, translation, periodic continuation limit process of scaling factor new quantities (effective parameters: porosity, permeability) new equations (porous media: instead of transport equations now Darcy-Forchheimer equation) Page 6 of 13 2.

7 3. 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 in 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 Page 7 of 13 2.

8 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-ɛ,... ) Page 8 of 13 2.

9 4. 2nd Example: Modelling of biofilm growth Discrete Models Lagrangian approach: microbes as moving particles within the fluid Eulerian approach, f.e. cellular automaton: (rectangular) cells filled 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 a microbes) Page 9 of 13 2.

10 How many dimensions? same resolution as for fluid dynamics? global concentration of microbes (same concentration everywhere) Choose resolution model any single microbe?? local groups of microbes for cellular automaton: just give states for microbes, or also concentrations? Multiscale and Hierarchical Modelling: resolve geometry (averaging over fine structures) In general: Population Modelling Page 10 of 13 2.

11 5. 3rd Example: Modeling Interactions Static or pseudo-static approach: Compute stationary flow field, and use result to simulate biofilm growth, or vice versa Compute stationary flow field, simulate biofilm growth during a small 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 each time step) fully coupled simulation: e.g.: extend system of differential equations to include all present effects Page 11 of 13 2.

12 6. Setting Up Simulation Experiments simulation has to be embedded input of data, visualization, feedback to modelling/design Example: biofilm modelling Page 12 of 13 2.

13 7. 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 exact solution (Simplex algorithm for linear programming) the approximative numerical approach: approximate/discretize the model equations and end up with some approximate solution Page 13 of 13 2.

CFD in COMSOL Multiphysics

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