Scientific Computing and Numerical Analysis Seminar October 1, 2010
Outline Heterogeneous Multiscale Method Adaptive Mesh ad Algorithm Refinement Equation-Free Method
Incorporates two scales (length, time or both) into one algorithm Different scales = different physical laws General idea behind CM methods: Utilize information from a detailed model to update information for a coarser model CM methods have been applied to a wide variety of applications in biology, chemistry, material science, and fluid mechanics
Heterogeneous Multiscale Method Developed by Wienan E and Bjorn Engquist Describe motion of a continuous body (e.g. fluid, gas, elastic material) whose motion is governed by its discrete elements (fluid particles) Modeling all discrete elements is too costly However, to close the continuum level equations, information is needed from the microscopic scale
Heterogeneous Multiscale Method Two Numerical Schemes Example: Modeling a particular fluid flow Continuum Level: Navier Stokes with Finite Volume Method Microscopic Level: Molecular Dynamics Equations for fluid particles Figure from http://www.answers.com/topic/laminar-flow and http://www.usp.edu/academics/collegesdepts/chemistry/
Heterogeneous Multiscale Method Numerical scheme chosen for discretizing and updating the continuum level equations (finite differencing, finite volume) Discretization corresponds to a grid laid over the domain For example, a one-dimensional, finite volume grid looks like this:
Heterogeneous Multiscale Method Example 1: Suppose you want to update some macroscopic (continuum-level) quantity U (velocity, density, etc.) U represents the average of u (the equivalent quantity at the microscopic scale) U i would represent the average fluid velocity or fluid density over the grid cell i U i (t) = 1 x x+ x 2 x x 2 u(y, t)dy (1)
Heterogeneous Multiscale Method Motion of quantities U or u governed by conservation laws of the form: U t + f (U) x = 0, u t + f (u) x = 0 The quantity changes over time due to any flux of the quantity into or out of an area The flux f (U) is often unknown or difficult to compute for macroscopic models It can be approximated by utilizing a known flux f (u) from the microscopic scale
Heterogeneous Multiscale Method Conservation Equation for the macroscopic U variable can be discretized as follows: U(x, t + t) U(x, t) f (u(x + x/2, t) f (u(x x/2, t) + t x where the flux of U is estimated by computing the flux of u exactly, at the boundaries of each grid cell
Heterogeneous Multiscale Method Example 2: Constitutive Law Modeling an elastic body motion Constitutive Law of the form: σ = σ(ɛ) is needed Law will contain parameters describing the material s response to mechanical strains For heterogeneous media, these values can vary with location in the material and with time HMM is used to update the mechanical parameters after each time step
Heterogeneous Multiscale Method Typical assumption made for this case: the microscopic elements quickly settle to equilibrium The computation of new mechanical parameters can be done with a small number of microscopic time steps The micro-system does not need to be evolved for the full macroscopic size time step
Heterogeneous Multiscale Method General Steps of HMM 1 Create a microscopic system based on macroscopic variables, (typically done with normal distributions) 2 Run the microscopic updating scheme 3 Apply an averaging operator to get macroscopic level values from the microscopic results 4 Run the macroscopic updating scheme 5 Repeat Steps 1-4
Heterogeneous Multiscale Method Pros and Cons of this Method: Pro: More accurate then just a continuous model Pro: Works well for problems with microscopic processes that settle quickly to equilibrium Con: HMM only works for problems where the micro-structure is well known, or can be reasonably approximated by a known distribution
Adaptive Mesh and Algorithm Refinement Developed mainly by Alejandro Garcia AMAR combines the ideas of grid refinement with utilizing different models at different refinement level Example: Navier-Stokes utilized at coarse grid, and a particle method used at the finest grid refinement
Adaptive Mesh and Algorithm Refinement Grid Refinement Lay a grid over the domain of the problem As simulation proceeds there may be regions in the domain that contain interesting dynamics Examples: near a boundary, in an area of turbulent flow The grid may need to be refined in these areas to get a better resolution of the solution
Adaptive Mesh and Algorithm Refinement Grid Refinement Refinement proceeds until error in grid values is below a certain threshold Can have different depths of refinements in different parts of the domain
Adaptive Mesh and Algorithm Refinement Grid Match-up It is important to have variables match-up at the boundaries of different refinement levels Averaging and Interpolation techniques are utilized
Adaptive Mesh and Algorithm Refinement If the refinement changes the spatial scale by several orders of magnitude, the equations may also change at this finer scale Example: Coarse grid, Navier Stokes Finest grid, Molecular Dynamics
Adaptive Mesh and Algorithm Refinement Basic Algorithm The continuous problem is advanced numerically one continuum-level time step t cont from t i to t i+1 over the whole coarse grid This is done even for coarse grid cells that overlay a finer grid The microscopic problem on the fine grid is also advanced from t i to t i+1 by taking several smaller time steps t micro
Adaptive Mesh and Algorithm Refinement Basic Algorithm (continued) The possible interaction of the refined area with its surrounding coarse grid cells is included by a "buffer" region at the boundaries between coarse and fine grids containing microscopic particles. These particles are moved during the microscopic advancement
Adaptive Mesh and Algorithm Refinement Basic Algorithm (continued) Any particles in the fine grid or buffer region that cross the boundary are included in the flux computation After all microsteps have been taken, the coarse grid cell overlying the fine grid has its variables updated by averaging the results from the fine scale computation
Adaptive Mesh and Algorithm Refinement Example Application: Piston traveling though a gas Shock wave forms as gas compressed Near shock, grid refined to gas particle level Navier-Stokes utilized at the coarse level AMAR captures the shock wave better than purely continuous Navier-Stokes
Adaptive Mesh and Algorithm Refinement Pros and Cons to AMAR Pro: Good for problems with small areas requiring refinement Pro: Microscopic problem advanced for full time interval so no microscopic information lost Con: Only useful for problems with small areas of interesting dynamics in the domain
Equation-Free Method Developed mainly by Yannis Kevrekidis EFM is similar to HMM in that the goal is to utilize micro-scale information to better model the system at the macro-scale Main difference: macro-scale equations are not explicitly written down and solved (hence the name "Equation-free!")
Equation-Free Method Motivation for EFM Sometimes the best description of a problem comes from a microscopic scale Developing a macroscopic scale, constitutive law for this system may be difficult EFM was developed to circumvent this issue Information from the micro-scale is used to estimate macro-scale variables at future points in time
Equation-Free Method Simple Example: Let C be some quantity (perhaps a concentration of a chemical) This quantity changes over time according to some law of the form: C = f (C) t Given the current state C n of the quantity, one can estimate C at the next time step with: C n+1 = C n + tf (C n )
Equation-Free Method Suppose the function f (C) is unknown However, suppose f (C) at several points in time is known: f (C 0 ), f (C 1 ),...f (C p ) These can be used in a numerical integration scheme to predict C at future points in time
Equation-Free Method EFM utilizes short bursts of microscopic computations to evolve microscopic variables c i, f (c i ) These variables are then averaged to compute the f (C i ) type values at the macro-scale The macroscopic f (C i ) s are then used in a numerical integration scheme to predict C at future points in time.
Equation-Free Method Algorithm Steps 1 Start with current values for the macroscopic variables 2 Create a microscopic system utilizing distribution functions based on the macroscopic variables 3 Run the computation at the micro-scale for a short number of steps 4 Average the micro-scale variables at each time step to convert them to macroscopic values 5 Estimate future macroscopic variables by utilizing a numerical time integration scheme
Equation-Free Method Applications Chemical Reactions Population Dynamics Disease Evolution
Equation-Free Method Pros and Cons Pro: Can be utilized for systems where continuum level equations are unknown or difficult to write down or solve Con: The microscopic data is not saved between macroscopic time steps so information can be lost
A Look Ahead... The downside to both HMM and EFM is that microscopic data is lost after each continuum time step My Continuum-Microscopic method seeks to correct this issue, by utilizing probability distribution functions to save micro-scale data over time This will be the topic of the next few seminar meetings
References E and Engquist, "Multiscale Modeling and Computation", (2003) Notices of the AMS, Vol 50, Number 9, p. 1062-1070 Garcia et al, "Adaptive Mesh and Algorithm Refinement using Direct Simulation Monte Carlo", (1999) Journal of Computational Physics, Vol 54, Issue 1, p. 134-155 Kevrekidis et al, "Equation-free: The computer-aided analysis of complex multsiscale systems", (2004), AIChE Journal, Vol 50, p. 1346-1355