We chose Euler's method for this introductory chapter because it is typical of many other classes of numerical methods. In essence, most consist of recasting mathematical opera- tions into the simple kind of algebraic and logical operations compatible with digital com- pllters. Figure 1.6 summarizes the major areas covered in this text.
P i p e 3 F l o w o u t : 1 2 0
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l 4 MATHEMATICAT MODELING, NUMERICAT METHODS, AND PROBLEM SOLVING TABTE l.l Devices ond types of bolonces fhot ore commonly used in ihe four moior oreos of engineering. For
eoch cose, lhe conservotion low on which the bolonce is bosed is specified.
Field Device OrganizingPrinciple MathematicalExpression
Chemical e n g i n e e r i n g
Civil e n g i n e e r i n g
M e c h a n i c a l e n g i n e e r i n g
Electrical e n g i n e e r i n g
tI
structyz ,,\
ffi \
?7fu. ,1m77,
ilH3
Conservation of mass
Conservation of momentum
Conservation o f m o m e n t u m
Conservation of charge
Conservation of energy
Force balance:
C u r r e n t b a l a n c e : + i , For each node
I current (i) = 0
I ' R '
Voltage balance:
a{A&-l , , R , - - 2 r * z YJ - - f
L--\A7\--J
i:R:
Around each loop
I emf's - I voltage drops for resistors
> 6 - > a : 0
Mass balance: ffi
inort
ff_--* ourpur
Over a unit of time period A m a s s : i n p u t s - o u t p u t s
+ F v
+I
- F n * O + + F H
VI
- a t /
At each node
I horizontal forces (FH) = o I vertical forces (I'u) : 0
Force balance: I Upward force
l r = 0I
V Downward forceI
m Li = downward force - upward force
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I.3 NUMERICAL METHODS COVERED IN THIS BOOK r 5
lal Part 2: Roots and optimization f(xl Roots: Solve for.r so thatfi-r) = 0
Optimization: Solve for x so that/'(r) = 0
lbl Part 3: Linear algebraic equations Given the a's and the b's. solve for the.r's f\x\
a ' , r x t t a r 2 x . a = b , arrx, 1- a,x, = b2
ldl Part 5: Integration and differentiation Integration: Find the area under the curve Differentiation: Find the slooe of the curve
lel Part 6: Differential equations G i v e n
dv Av
, h :
N : f l t ' Y l
solve for r as a function of r .,Ii+r = -]'i + "f(ti, yJAr
F I G U R E I . 6
Summory of the numericol methods covered in this book.
Optima
lcl Part 4: Curve fitting
o
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t 6 MATHEMATICAL MODELING, NUMERICAT METHODS, AND PROBLEM SOLVING Part Two deals with two related topics: root finding and optimization. As depicted in Fig. 1.6a, root locotiorr involves searching for the zeros of a function. In contrast, optimiza- rion involves determining a value or values of an independent variable that correspond to a
"best" or optirnal value of a function. Thus, as in Fig. 1 .6a, optimization involves identify- ing maximir and minima. Although somewhat different approaches are used, root location and optimization both typically arise in design contexts.
Part Three is devoted to solving systems of simultaneous linear algebraic equations (Fig. 1.6&). Such systerns are similar in spirit to l'oots of equations in the sense that they are concemed with values that satisfy equations. However, in contrast to satistying a single equatiou, a set of values is sought that simultaneously satisfies a set of linear algebraic equations. Such equations arise in a variety of problem contexts and in all disciplines of en- gineeriug and science. In particular, they originate in the mathenratical modeling of Jarge systems of interconnected elements such as structures, electric circuits. and fluid networks.
However, they are also encountered in other areas of numerical methods such as curve tit- l i n g l r n d d i f f e r e n t i a l e q u u t i o n s .
As an engineer or scientist. you will often have occasion to fit curves to data points. The techniques developed for this pulpose can be divided into two general categories: regression and interpolation. As described in Part Four tFig. 1.6c'1, regression is ernployed where there is a significant degree of error associirted with the data. Experimental results are often of this kind. For these situations. the strategy is to derive a single curve that represents the general trend of the data without necessarily matching any individual points.
In contrast, interpolution is used where the objective is to determine intermediate val- ues between relatively error-free data points. Such is usually the case for tabulated infor- mation. The strategy in such cases is to flt a curve directly through the data points and r.rse the curve to predict the intermediate values.
As depicted in Fig. 1.6d, Part Five is devoted to integlation and differentiation. A plrysical interpretation of ruurrcricctl iltegratiott is tlre determination of the area under a curve. Integration has many applications in engineering and science, ranging from the de- termination of the centroids of oddly shaped objects to the calculation of total quantities based on sets of discrete measurements. In addition, nurnerical integration formulas play an importtrnt role in the solution of diffbrential equations. Part Five also covers methods for nume.rical difr'erentiation. As you know fiom your study of calculus, this involves the de- termination of a function's slope or its rate of change.
Finally. Part Si.x focuses on the solution of ordirro'v di.fterential equations (Fig. 1.6e).
Such equations are of great significance in all areirs of engineering and science. This is be- cause many physical laws are couched in terms of the rate of change of a quantity rather than the magnitude of the quantity itself. Examples range from population-forecasting rnodels (rate ofchange of population) to tlre acceleration of a tallin-e body (rate ofchange ofvelocity).
Two types of problems are addressed: initial-value and boundary-value problems.
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PROBLEMS t 7