ENGM ENGM 541, ENGM 670 541, ENGM 670-XX55 && MECE 758MECE...
Transcript of ENGM ENGM 541, ENGM 670 541, ENGM 670-XX55 && MECE 758MECE...
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ENGM ENGM 541, ENGM 670541, ENGM 670--X5X5
& & MECE 758MECE 758--X5X5Modeling and Simulation of Engineering SystemsModeling and Simulation of Engineering Systems
Winter Winter 20112011
Lecture 2:Lecture 2:
ExtremumExtremum Approach; Approach;
General Formulation of General Formulation of
LumpedLumped--Parameter Equilibrium Systems;Parameter Equilibrium Systems;
Solution MethodsSolution Methods
M.G. LipsettM.G. Lipsett
Department of Mechanical EngineeringDepartment of Mechanical Engineering
University of AlbertaUniversity of Albertahttp://www.ualberta.ca/~mlipsett/ENGM541/ENGM541.htm
© MG Lipsett, 2011 2
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Review of Modeling & SimulationReview of Modeling & Simulation
• A model is a representation of knowledge
– Rules, physical analogs, algebraic equations of physical laws
• A system is a bounded region comprising known elements
that each interact in understandable ways
• The types of systems of interest in this course include:
– Models of physical systems
• Mechanical, electrical, thermal, structural, hydraulic, etc.
– Models of material, energy, and information flow for engineering
decisions
• Production systems, Economics, Scheduling, Inventory, …
• Simulations are solutions of differential equations that are
functions of time
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Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Equilibrium Problems for LumpedEquilibrium Problems for Lumped--Parameter SystemsParameter Systems
• Steady-state solutions (no change over time)
• In a lumped-parameter problem, the continuous system has been modeled using systems of interconnected elementselementsconnected at nodesnodes
•• LoopLoop variables describe the path around the loop of some
or all of the elements
Node variables describe variables that go through an
element and which come together at a node.
The loop and node variables are related by the constitutive
relationships of the elements.
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Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Formulating Constitutive RelationshipsFormulating Constitutive Relationships
1. State the variables
2. Describe the element
3. Sketch the constitutive relationship.
4. Use an analytic expression for the relationship
Loop variable
Node variable
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Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
General Procedure for Setting Up A ProblemGeneral Procedure for Setting Up A Problem
1. Choose the variable in which you want your final equations expressed
2. Choose variables so as to satisfy the pertinent admissibility requirement
3. Choose other variable type & write as many equations as necessary to check that admissibility is satisfied.
4. Relate the loop and node variables using the constitutive relationships.
5. Eliminate all but the chosen variables (all of one type) from the equations). Substitute in the equations and group terms.
6. Non-dimensionalise the variables.
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Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Extremum FunctionsExtremum Functions
• The other way of formulating the equations governing systems is to use extremum functions. This includes energy methods.
• We make up a scalar function from the constitutive relationships of all the elements in the system, and search for an extreme value of the function (e.g. minimum
potential energy).
• We go back to our original definition of a constitutive
relationship to define two quantities:
1. Content U (energy)
2. Co-Content U* (co-energy)
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© MG Lipsett, 2011 7
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
EnergyEnergy
• Area under the curve is the energy U in the element:
• We write p (which is a node variable) as a function of q(loop variable) and U becomes a function of q only.
© MG Lipsett, 2011 8
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
CoCo--EnergyEnergy
• Similarly to energy, with co-energy U* as a function of p only
• For all sets of state variables satisfying node (loop) admissibility, those also satisfying loop (node) admissibility will render the co-energy (energy) an extreme value.
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© MG Lipsett, 2011 9
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example: Extremum ApproachExample: Extremum Approach
Recall our system of carts and springs.
There are two types of elements in this system:
1) Spring
2) Force source (which yields a “through” variable)
K/6
K/6
K/3 K/2
2P P
© MG Lipsett, 2011 10
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (2): Spring Constitutive RelationshipExample (2): Spring Constitutive Relationship
• Consider the spring constitutive relationship:
• Force f through distance δ does positive work on the spring,
δ
loop variable
node variable
f
f = k δ w.r.t loop variable; or
δ = (1/k) f w.r.t node variable
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© MG Lipsett, 2011 11
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (3): Energy & CoExample (3): Energy & Co--Energy for SpringEnergy for Spring
• Energy:
• Co-Energy:
© MG Lipsett, 2011 12
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (4): Force SourceExample (4): Force Source
• A specified force regardless of the deflection, visualised as:
• Like a potential energy term for a hanging mass
• Once again we must be careful about signs
• Question: What’s U* for a force source?
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Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (5):Displacement SourceExample (5):Displacement Source
• In other problems, we might have a displacement source,
which is a prescribed displacement regardless of the force
applied – physical constraint such as a stop, a track, or a
wall. In other types of systems this would be analogous to a
voltage source, a temperature source, or a pressure source.
• The constitutive relationship for a displacement source is
visualised as follows:
© MG Lipsett, 2011 14
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (6): Example (6): Spring Force Equations & CoSpring Force Equations & Co--EnergyEnergy
1) Choose an admissible set of node variables (forces at nodes).
To be admissible, the variables need to satisfy the node
admissibility requirements:
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© MG Lipsett, 2011 15
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (7): CoExample (7): Co--Energy ExpressionEnergy Expression
• Now we write an expression for the co-energy of the system,
with one term for each element in the system:
• For solution, U* will have an extreme value:
• We first differentiate U* by f1 :
© MG Lipsett, 2011 16
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (7): NonExample (7): Non--DimensionaliseDimensionalise
• To non-dimensionalise:
• And get
• Next, we differentiate U* by f2 to get
• And put the two expressions into matrix form as
• Recall from the direct formulation that the solution is
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Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example (8): Symmetric Matrix FormExample (8): Symmetric Matrix Form
• Note that the coefficient matrix resultant from an extremum
function approach is symmetricsymmetric.
• This is always true.
• A symmetric coefficient matrix is a good check for the
solution (but it doesn’t guarantee the right answer!)
© MG Lipsett, 2011 18
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example #2: Fluid Flowing in PipesExample #2: Fluid Flowing in Pipes
Given: P0, P4
Elements are pipe sections
and pressure sources
Consider the constitutive relationship for a pipe section:
P0
P4P4
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© MG Lipsett, 2011 19
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example #2 (2): Pressure SourceExample #2 (2): Pressure Source
∆ P = Pi – Pj = constant (regardless of flow q)
Co-energy = q ( Pi – Pj )
Now, construct in terms of flows in the pipe segments
(elements) a node variable formulation using co-energy:
Pi Pj
© MG Lipsett, 2011 20
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example #2 (3): Define Admissible Node Variables Example #2 (3): Define Admissible Node Variables
• Node 1:
• Node 2:
• Node 3:
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© MG Lipsett, 2011 21
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example #2 (4): Write Total CoExample #2 (4): Write Total Co--Energy of SystemEnergy of System
To eliminate variables, express them in terms of other variables
Write the total co-energy of the system:
Substitute for q1 , q5 , & q6 (from node admissibility):
© MG Lipsett, 2011 22
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example #2 (5): Find Extreme ValueExample #2 (5): Find Extreme Value
For solution, U* has extreme value, so we generate a set of three equations (one for each admissible variable):
And solve to get
where
=
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© MG Lipsett, 2011 23
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Formulating the General Equilibrium ProblemFormulating the General Equilibrium Problem
• Any formulation we develop will start with a set of variables
xi , i = 1, …, N ; they represent the state of the system. We
determine N functions of xi , each equal to some forcing
value Ci.
• If any of the xi appears to a power other than 1, or is combined in a multiplicative sense with any other xi , then the entire system is nonlinear.
• Very often, however, we find that this is not the case and that the equations are linear.
© MG Lipsett, 2011 24
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Review of Form of Linear EquationsReview of Form of Linear Equations
Linear equations can be written as:
which we can write in matrix form as
known, constant coefficient matrix knowns
We typically write this as
The solution will often be written as:
=
=
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© MG Lipsett, 2011 25
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Solutions to Linear EquationsSolutions to Linear Equations
• This notation is usually restricted to analytical work.
Numerical solutions may or may not invert the matrix,
depending on the efficiency.
• An efficient algorithm will minimise the number of
operations &/or the storage requirements.
• We will consider different methods for solving [A]{x} = {c}
Cramer’s Rule: Use Determinants
• If we want to find xi , we find the determinant of the coefficient matrix, then we replace the i th column of [A] with the vector {c}, find another determinant, and divide it by the first determinant.
© MG Lipsett, 2011 26
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Example: Determinants to Solve for {Example: Determinants to Solve for {xx}}
Say we have the system
and we want to find x2.
=
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© MG Lipsett, 2011 27
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Computational RequirementsComputational Requirements
The computational effort (number of
operations, counting multiplications &
divisions, not additions & subtractions)
involved in taking the determinant
grows as N! (N-factorial), which is the
fastest growing function known.
N N !
1 1
2 2
3 6
4 24
5 120
… …
10 3,628,800
… …
20 2.43 x 1018
© MG Lipsett, 2011 28
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Gaussian Elimination Method for Solving {Gaussian Elimination Method for Solving {xx}}
The most familiar method, comprising two parts:
An elimination phase to transform [A]{x} = {c} into [U]{x}={d}
where
followed by a solution phase to find xi , i = 1, …, N
by back substitution
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© MG Lipsett, 2011 29
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Gaussian Elimination Method (2)Gaussian Elimination Method (2)
• In the first phase, each row (equation) is first divided by its first element, and then the first equation is subtracted from
all the other equations, leaving us with
• Zeroing the first column requires ~ N2 operations. We
eventually want to get all zeros below the diagonal.
• We repeat this process again using the remaining elements in the lower right (N-1)x(N-1) submatrix of [A] to give
=
=
© MG Lipsett, 2011 30
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Gaussian Elimination Method (3)Gaussian Elimination Method (3)
After decomposing N rows we get the upper diagonal matrix [U]. From
• We find xN , then back substitute to get xN-1 , & so on, & so on.
• The overall number of operations varies on the order of N3
N N3
10 1000
… …
100 106
Size of system Number of operations
(which is still significantly smaller than for Cramer’s Rule when N <6)
[ ]
=
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© MG Lipsett, 2011 31
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Systems of Equations with Banded FormSystems of Equations with Banded Form
• We often find that our equations have a banded form, which
means that all of the non-zero coefficients are near the
diagonal, and all the terms far away from the diagonal are
zero.
• If we do Gaussian elimination on this banded matrix we only need M2 operations; total number of operations ≈ M2N .
© MG Lipsett, 2011 32
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
LimitedLimited--Bandwidth Equilibrium ProblemsBandwidth Equilibrium Problems
• Coefficient matrices with limited bandwidth are very common in engineering systems, and typically arise when a node is only acted upon by its nearest neighbouring nodes
through the neighbouring elements.
• We can defeat the advantages of bandwidth if we choose an inappropriate numbering system for the variables of the
problem.
• Example: We want to find the displacements of all the carts.
• Choose unique (admissible) loop variables u1, u2 , u3 , u4 , u5
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© MG Lipsett, 2011 33
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
LimitedLimited--Bandwidth Equilibrium Problems (2)Bandwidth Equilibrium Problems (2)
Node 1:
Node 2:
Etc. …, yielding
=
© MG Lipsett, 2011 34
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
LimitedLimited--Bandwidth Equilibrium Problems (3)Bandwidth Equilibrium Problems (3)
• If, on the other hand, this were the case:
• Then the non-zero elements of the coefficient matrix would look like this:
• Which is a full matrix, and so we can’t use the more efficient limited bandwidth solution. (If we had numbered u2 as u5, or u5 as u2, then we would have ended up with a full matrix as well. Try to number sequentially.)
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© MG Lipsett, 2011 35
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Another Example of Node NumberingAnother Example of Node Numbering
• Steady-state heat conduction into a hollow cylinder, considered as a two-dimensional problem
• We assume a set of nodes, each with a unique temperature. This satisfies loop admissibility automatically.
• The number of unknowns equals the number of nodes.
• Notice there is some symmetry. The system is symmetric about the vertical axis through its centre, so we only have to analyse half the cylinder.
Equilibrium problem: what is the
temperature distribution in the wall?
© MG Lipsett, 2011 36
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Node Numbering Example (2)Node Numbering Example (2)
• Decide on a temperature profile of interest, say radial lines 10° apart (19 lines: 0° to 180°) with 10 nodes within the wall on each radial line
• Total of 190 nodes & 190 unknown temperatures
• The problem could thus be formulated as
• If we use a random numbering for the nodes, then we could fill the matrix. In that case, the effort to solve using Gaussian elimination would be ≈ (190)3 ≈ 8 x 106 operations
=
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© MG Lipsett, 2011 37
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Node Numbering Example (3)Node Numbering Example (3)
• Consider a couple of radial lines and choose an efficient
numbering scheme to reduce the bandwidth of the problem
BC
D E
A F
0°
10°20°
30°
G
We have a couple of options:
•Proceed from A along circumference along half-circle;
•Return along next higher radius set of nodes;
•Include next radius node out for connection point.
Bandwidth is 19 + 19 + 1 = 39
Or:
•Proceed along radial line from A to D;
•Return along neighbouring radius to F;
•Index to include G.
Bandwidth is 10 + 10 + 1 = 21. Effort in solving is
M2N ≈ (21)2(190) ≈ 8 x 104, which is 100 x more efficient than
solving the full matrix
© MG Lipsett, 2011 38
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Matrix InversionMatrix Inversion
• Solving the system of linear equations [A]{x} = {c} by inverting the matrix yields
• which involves a matrix inversion and a matrix multiplication. We can think of matrix inversion as a set of Nequilibrium problems, in which each column of [A]-1 is a
vector of unknowns & each column of [I] is a different right-hand side:
• So there are N such problems. Matrix inversion is only efficient if solving significantly more than N problems with
the same coefficient matrix.
[ ] { } { }=
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© MG Lipsett, 2011 39
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Matrix InversionMatrix Inversion
• Solving the system of linear equations [A]{x} = {c} by
inverting the matrix yields
• which involves a matrix inversion and a matrix
multiplication. We can think of matrix inversion as a set of N
equilibrium problems, in which each column of [A]-1 is a
vector of unknowns & each column of [I] is a different right-
hand side: Expanding, this becomes
• So there are N such problems. Matrix inversion is only efficient if solving significantly more than N problems with the same coefficient matrix.
[ ]
=
[ ] { } { }=
[ ]
=
[ ] { } { }=
[ ] { } { }=
© MG Lipsett, 2011 40
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Iterative TechniquesIterative Techniques
• Iterative solution methods are indirect, theoretically taking an infinite number of steps to find a solution from an initial guess.
• Practically, they will be “close enough” after a finite number of steps; but we can’t predict exactly the number of steps required to get close enough.
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© MG Lipsett, 2011 41
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Iteration by Total StepsIteration by Total Steps
• Consider the following expression:
(which has N equations of this type)
• We start by picking an initial set of guesses for the
unknowns
• When we substitute these guesses into the equations, typically they won’t provide a satisfactory solution
• So we need to refine the set of guesses
© MG Lipsett, 2011 42
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Iteration by Total Steps (2)Iteration by Total Steps (2)
• We can refine the set of guesses to find a new (and hopefully improved) set
• using the equations themselves, according to
Example: Given the set of equations in three unknowns:
Solve for x1, x2, and x3
=
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© MG Lipsett, 2011 43
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Method of Iteration by Total StepsMethod of Iteration by Total Steps
• Assume an initial set of guesses (and then at each
successive step use the most recent guesses):
• Having calculated for x1(1), x2
(1), and x3(1), we check the
equations.
• If they are not satisfied, we repeat, using the most recent guesses x1
(1), x2(1), and x3
(1) to generate another set of guesses x1
(2), x2(2), and x3
(2)
© MG Lipsett, 2011 44
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Notes on Total Steps Calculation MethodNotes on Total Steps Calculation Method• The calculation method takes all the terms but the diagonal on to the
RHS, then divides by the diagonal coefficient to get an expression for
the variable associated with this row.
•• ConvergenceConvergence is met when the values of the set of guesses changes
very little after a step. We can check in advance whether we can expect
convergence using a row-sum criterion:
• Add up the absolute values of the coefficients in a row of [A]
• Divide the sum by the absolute value of the diagonal element
• Generate the terms
• Do a test to see whether all
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© MG Lipsett, 2011 45
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
SummarySummary
• We have developed methods for generating governing
equations for equilibrium systems using extrema.
• We have seen a few of the many methods for the solving
the equations for linear systems.
Next lecture:
• We will look at other iterative solution methods for (big)
linear systems, and then consider how to deal with
nonlinearities.
• We will also look at what kinds of equilibrium systems arise in engineering.
• Then we will move to modeling time-varying systems.
© MG Lipsett, 2011 46
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, 670-X5, MECE 758 – Modeling and Simulation of Engineering Systems
Lecture 2: Extremum Approach; General Formulation for Lumped-Parameter Equilibrium
Break Time: Optical IllusionsBreak Time: Optical Illusions
Source: http://www.mcescher.com
M.C. Escher breaks the loop admissibility law.