Modeling & simulation

Ananya K B - PR13CE2002

Jisha John - PR13CE2006

MODELING

Process of producing a modelModel is the representation of construction and

working of a system of interestModel will be similar but simpler than the

systemA good model is a judicious trade off between

realism and simplicity

Modeling processPhase i: step 1- data collection step 2- model input preparation step 3 - parameter evaluationPhase ii: step 4 - calibration step 5- validation step 6 - post auditPhase iii: step 7 - analysis of alternatives

Code verified?

Define Purpose

Field data Conceptual Model

Mathematical Model

Analytical Solutions

Numerical formulation

Computer program

no

CODE SELECTION yes

Model design Field data

calibrationComparison with field data

verification

prediction

Presentation of results

PostauditField data

Types of models4 types of models in modeling 1. mathematical models 2. conceptual models 3. physical models 4. computational models 5. graphical model.

MATHEMATICAL MODELS• these are simplified representations of some

real world entity• These can be in equations or computer code• These are intended to mimic essential features

while leaving out inessentials• Assumptions in mathematical modeling 1. variables ( the things which change) 2. parameters (the things which do not change) 3. functional forms(the relationship between

two)

MATHEMATICAL MODEL

• Use symbolic notation and mathematical equations to represent a system. Attributes are represented by variables and the activities represented by mathematical functions that interrelate the variables• simulates ground-water flow and/or solute fate

and transport indirectly by means of a set of governing equations thought to represent the physical processes that occur in the system.

Components of a Mathematical Model

• Governing Equation

(Darcy’s law + water balance equation) with head (h) as the dependent variable • Boundary Conditions• Initial conditions (for transient problems)

Darcy’s Law

If the soil did not have uniform properties, then we would have to use the continuous form of the derivative:Q(x)= -K(x) * A* dH / dx

Head is defined as the elevation to which ground water will rise in a cased well. Mathematically, head (h) is expressed by the following equation:wherez = elevation head andP/pg = pressure head (water table = 0).

Types of Boundary Conditions

Specified Head Boundaries

Specified Flux Boundaries

Head Dependant Flux Boundaries

Specified Head BoundariesBoundaries along which the heads have been measured and

can be specified in the modele.g., surface water bodies

They must be in good hydraulic connection with the aquiferMust influence heads throughout layer being modeledLarge streams and lakes in unconfined aquifers with highly permeable beds

Uniform Head Boundaries: Head is uniform in space, e.g., Lakes

Spatially Varying Head Boundaries: e.g., River heads can be picked of of a topo map if:

Hydraulic connection with and unconfined aquiferthe streambed materials are more permeable than the aquifer materials

Specified Flux Boundaries:

Boundaries along which, or cells within which, inflows or outflows are set

Recharge due to infiltration (R)Pumping wells (Qp)Induced infiltrationUnderflowNo flow boundaries

Valley wall of low permeable sediment or rockFault

Need of mathematical modeling

1. Scientific understanding2. Clarification3. Manage the world using scientific

understanding4. Simulated experimentation5. The curse of dimensionality

Scientific understanding

• A model embodies a hypothesis about the study system, and lets you compare that hypothesis with data.

• A model is often most useful when it fails to fit the data, because that says that some of your ideas about the study system are wrong.

• Mathematical models and computer simulations are useful experimental tools for building and testing theories, assessing quantitative conjectures, answering specific questions, determining sensitivities to changes in parameter values and estimating key parameters from data.

CLARIFICATION

• The model formulation process clarifies assumptions, variables, and parameters

• The process of formulating an ecological model is extremely helpful for organizing one’s thinking, bringing hidden assumptions to light and identifying data needs

Manage the real world using scientific understanding

• Forecasting disease or pest outbreaks• Designing man-made systems, for example,

biological pest control, bioengineering• Managing existing systems such as agriculture or

fisheries• Optimizing medical treatments

Simulated experimentation

Realistic experimenting may be impossible• Experiments with infectious disease spread in

human populations are often impossible, unethical or expensive.

• We can not manage endangered species by trial and error.

• We dare not set dosage for clinical trials of new drugs on humans or set safe limits for exposure to toxic substances without proper knowledge of the consequences.

The curse of dimensionality

• Sometimes a purely experimental approach is not feasible because the data requirements for estimating a model grow rapidly in the number of variables.

• Modelling using computer programs is cheap

Types of mathematical model

• Deterministic vs. Stochastic models• Static vs. Dynamic Models• Continuous vs. Discrete Models• Individual vs. Structured Models• Mechanistic vs. Statistical Models• Qualitative vs. Quantitative Models

Conceptual Model

• Qualitative description of the system

A descriptive representation of a groundwater system that incorporates an interpretation of the geological & hydrological conditions.

Graphical Model

• FLOW NETS– limited to steady state, homogeneous systems,

with simple boundary conditions

PHYSICAL MODELSystem representation by “physical means” “Electrical, mechanical, hydraulic” or other physical representation of the system”. For example , in a physical model of a system , if the system attributes can be represented by such measurements as a voltage, then the rate at which the shaft of a direct current motor turns depends upon the voltage applied to the motor.• SAND TANK

– which poses scaling problems, for example the grains of a scaled down sand tank model are on the order of the size of a house in the system being simulated

SAND MODEL

• Calibration of model : Test of the model with known input and output information that is used to adjust or estimate factors for which data are not available.

• Verification of the model: examination of the numerical technique in the computer code to ascertain that in truly represents the conceptual model and that there are no inherent numerical problems with obtaining a solution

• Validation: comparison of model results with numerical data independently derived from experiments or observations of the environment.

COMPUTATIONAL MODELLING

• Computational modelling is the use of mathematics, physics and computer science to study the behaviour of complex systems by computer simulation. A computational model contains numerous variables that characterize the system being studied. Simulation is done by adjusting these variables and observing how the changes affect the outcomes predicted by the model. The results of model simulations help researchers make predictions about what will happen in the real system that is being studied in response to changing conditions.

SIMULATION• Operation of a model of the system• It is a tool to evaluate the performance of a

system, existing or proposed, under different configurations of interest and over long periods of real time.

SIMULATION IS USED……

1.For built a new system or to alter an existing system

2. To reduce the chances of failure to meet specifications

3. To eliminate unforeseen bottlenecks4. To prevent under or over-utilization of

resources5. Optimize system performance

SIMULATION STUDY

Steps involved in developing a simulation modelIdentify the problemFormulate the problemCollect and process real system dataFormulate and develop modelValidate the modelDocument model for future useSelect appropriate experimental designEstablish experimental conditions for runsPerform simulation runsInterpret and present resultsRecommend further course of action

Identify the problem : find out the problems with an existing system. Produce requirements for a proposed system

Formulate the problem: objective of the study or issues involved in the system

Collect and process real system data: collect the data on system like input variables, performance of the system etc. identify the stochastic input variables.

Formulate and develop a model: develop schematics and network diagrams of the system. Translate these conceptual models to simulation software acceptable form.

Validate the model: compare the performance of the model under known conditions with the performance of the real system.

Document objectives, assumptions and input variables of model in detail for future use.

Select appropriate experimental design: maximum performance with minimum number of inputs

Establish experimental conditions for runs: run each model and find out the best performance from the run

Advantages of simulation It is useful for sensitivity analysis of complex systems.

It is suitable to analyze large and complex real life

problems that cannot be solved by the usual

quantitative methods.

It is the remaining tool when all other techniques

become intractable or fail.

It can be used as a pre-service test to try out new

policies and decision rules for operating a system.

Disadvantages of simulation Sometimes simulation models are expensive and take a long

time to develop.

Each application of simulation is ad hoc to a great extent.

The simulation model does not produce answers by itself.

It is the trial and error approach that produces different

solutions in repeated runs .It does not generate optimal

solutions to the problems.

PARAMETER ESTIMATION• In mathematical model all quantities are measured as

parameters, so this model is a parametric function.

• Inferring parameters from measurements is known as estimation

• 2 types of estimation 1. Parametric estimation where the quantities to be estimated are the unknown variables in equations that express the observables 2. Condition estimation where conditions can be formulated among the observations. Rarely used, most common application is levelling where the sum of the height differences around closed circuits must be zero

Steps in parametric estimationObservation equations: equations that relate the

parameters to be estimated to the observed quantities

Stochastic model: Statistical description that describes the random fluctuations in the measurements

Inversion that determines the parameters values from the mathematical model consistent with the statistical model.

Sensitivity analysis• Sensitivity analysis is the study of how

the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be apportioned to different sources of uncertainty in its inputs.

A sensitivity analysis can be used to

• validate a model,• warn of unrealistic model behavior,• point out important assumptions,• help formulate model structure,• simplify a model,• suggest new experiments,• guide future data collection efforts,• suggest accuracy for calculating parameters,• adjust numerical values of parameters,• choose an operating point,• allocate resources,• detect critical criteria,• suggest manufacturing tolerances, • identify cost drivers.

In a sensitivity analysischange

• the values of • inputs• parameters

• architectural featuresmeasure changes in

• outputs• performance indices

Conceptual models • type of diagram which shows of a set of relationshi

ps between factors that are believed to impact or lead to a target condition; a diagram that defines theoretical entities, objects, or conditions of a system and the relationships between them

A complete conceptual model provides a:• Definition of the phenomenon in terms of features

recognizable by observations, analysis or validated simulations;

• Description of its life cycle in terms of appearance, size, intensity and accompanying weather;

• Statement of the controlling physical processes which enables the understanding of the factors that determine the mode and rate of evolution of the phenomenon;

• Specification of the key meteorological fields demonstrating the main processes;

• Guidance for predicted meteorological conditions or situations using the diagnostic and prognostic fields that best discriminate between development or non-development; guidance for predicting displacement and evolution.

Advantageous of conceptual model• When introducing a new topic in class regardless of

whether the ultimate goal is to develop the topic qualitatively or quantitatively.

• When equations for some process being studied seem to obscure student understanding it is a good idea to step back a bit and discuss a conceptual model of the processes. Actually, it is typically best to develop a conceptual framework for understanding before introducing equations.

• To help explain and discuss interesting features in data sets.