Thompson, Warsi, Mastin - Numerical Grid Generation, страница 41

This occurs because the grid ceases to move when the measure isuniform, i.e., when the local value is equal to the average value everywhere. Therefore, thegrid can be considered to move so as to minimize the variation in the measure over the field.B. Reaction analogyA different, but somewhat related, approach was noted in Ref. [45] and [5] based on achemical reaction analogy. Here each grid interval is taken to represent a speciesconcentration, and the reaction rate constants are made dependent on the difference betweena local error measure for one grid interval compared with another.

Each grid interval then iscoupled with every other grid interval through reaction rate equations, so that each intervalgrows at the expense of others, and vice versa. A system of ordinary differential equations issolved for the intervals. This approach, as given, is somewhat inefficient, since there is noprovision for limiting the effect to the nearer points.

With each point affected equally by allother points, the number of ordinary differential equations to be solved is equal to the squareof the total number of points.The rate constants also contain factors designed to limit the range of variation of thegrid intervals. The two-dimensional form given involves essentially applying theone-dimensional form separately along each family of curvilinear coordinate lines, withspacing in one cartesian coordinate being adjusted along one family of curvilinear lines, andthe other cartesian coordinate being adjusted along the other family.C. Moving finite elementsThe moving finite element method of Miller (Ref.

[49] -- [50]) is adynamically-adaptive finite element grid method in which the grid point locations are madeadditional dependent variables in a Galerkin formulation. The solution is expanded inpiecewise linear functions, in terms of its values at the grid points and those of the grid pointlocations on each element. The residual is then required to be orthogonal to all the basisfunctions for both the solution and the grid. The grid point locations are thus obtained as partof the finite element solution.

An internodal viscosity is introduced to penalize the relativemotion between the grid points. This does not penalize the absolute motion of the points. Aninternodal repulsive force was also introduced to maintain a minimum point separation. Bothof these effects are strong but of short range. A small long range attractive force is alsointroduced to keep the nodes more equally spaced in the absence of solution gradients.

Smalltime steps are used in the initial development of the solution. The results show that theoscillations typically associated with shocks with fixed grids are removed with the adaptivegrid, and that dispersion and dissipation are essentially eliminated. An order-of-magnitudeincrease in stability was also realized over conventional methods.5.

CorrelationsThe ultimate answer to numerical solution of partial differential equations may well bedynamically-adaptive grids, rather than more elaborate difference representations andsolution methods. It has been noted by several authors that when the grid is right, mostnumerical solution methods work well.

Oscillations associated with cell Reynolds numberand with shocks in fluid mechanics computations have been shown to be eliminated withadaptive grids. Even the numerical viscosity introduced by upwind differencing is reduced asthe grid adapts to regions of large solution variation. The results have clearly indicated thataccurate numerical solutions can be obtained when the grid points are properly located.It is also clear that there is considerable commonality among the various approaches toadaptive grids.

All are essentially variational methods for the extremization of some solutionproperty. The explicit use of varational principles allows effective control to be exercisedover the conflicting requirements of smoothness, orthogonality, and concentration, and thisis probably the most promising approach in multiple dimensions.The adaptive grid is most effective when it is dynamically coupled with the physicalsolution, so that the solution and the grid are solved for together in a single continuousproblem.

The most fruitful directions for future effort thus are probably in the developmentand direct application of variational principles and in intimate coupling of the grid with thephysical solution.Exercises1. Show that Eq. (4) is the Euler equation for the minimization of the integrals (5), (6), (12),and (13). Hint: For (6) note that in the term, w must be differentiated withrespect to implicity, i.e., w =wxx . A similar situation occurs with (13). Note, however,that implicit differentiation is not to be used in the termfor (5) or infor (12).2. Show that Eq. (4) is also the Euler equation for the integrals3.

With the weight function given by w(x)=sin( x/L), find the grid point locations from Eq.(20). Note the concentration near x=L/2 where the weight function has its maximum value.4. For u(x)=(L/ )sin( x/L), obtain the point distribution from Eq. (20) using the weightfunctions from Eq.

(22), (23), (25), (26) and (27). Use = =1. Plot and compare.5. Show that the average of the squares of the diagonal lengths is.6. Verify the correspondence between the six grid properties listed on p. 395 with the sixlisted on p. 396. Hint: Recall that dx= d .7.

Verify that the one-dimensional forms of the first four properties on pp. 395-396 are asstated on p. 396-397. Hint: In one dimension take8. Show that Eq. (59) is the Euler equation resulting from the integral given by Eq. (57).9. Verify Eq. (72) and (73).10. Show that withthe generation system isHint: Use Eq. (19) of Appendix B.11. Show that within the computational space.the generation system consists of Laplace equations12. Show that withthe generation system is13. Show that withHint: Note that, (i,j,k) cyclic, the generation system isand.APPENDIX ADIFFERENTIAL-GEOMETRIC CONCEPTS ON SPACE CURVES AND SURFACES1.

Theory of CurvesIn this appendix we consider only those parts of the theory of curves in space whichare needed in the theory of surface geometry for the purpose of coordinate generation. Let Cbe a curve in space whose parametric equation is given aswhere is a parameter which takes values in a certain interval ab.It is assumed that the real vector function ( ) is p 1 times continuously differentiable forall values of in the specified interval, and at least one component of the first derivativeis different from zero. Note that the parameter can be replaced by some other parameter,say s, provided that ds/d0.A. Tangent vectorLet us consider the arc length s as a parameter.

Then the coordinates of twoneighboring points on the curve are (s) and (s+h). The vector (s) defined as(3)is the unit tangent vector at the point s on the curve. Sincethat, we immediately see.If the curve C is referred to a general coordinate systemequations are given asi,then its parametricIn this case, using the chain rule of differentiation, we can write(4)wherei arethe covariant base vectors defined in Eq. (III-1).B. Principal normalSince, a single differentiation with respect to s yieldsso that the vector d /ds is orthogonal to. The vector(5)is called the curvature vector.

The unit principal normal vector is then defined as(6)and its reciprocal = 1/k(s) are, respectively, theThe magnitudecurvature and the radius of curvature of the curve at the point under consideration. Both thecurvature vector and the principal normal are directed toward the center of curvature of thecurve at that point.C. Normal and osculating planesThe totality of all vectors which are bound at a point of the curve and which areorthogonal to the unit tangent vector at that point lie in a plane. This plane is called thenormal plane. The plane formed by the unit tangent and the principal normal vector is calledthe osculating plane.D. Binormal vectorA unit vector (s) which is orthogonal to both and is called the binormal vector.Its orientation is fixed by taking , , to form a right-handed triad as shown below:(7)Note that for plane curves the binormal is the constant unit vector normal to the plane,and the principal normal is the usual normal to the curve directed toward the center ofcurvature at that point.The twisted curves in space have their binormals as functions of s.

Because of twistinga new quantity called torsion appears, which is obtained as follows. Consider the obviousequations(8)Differentiating each equation with respect to s, we obtain(9a)(9b)Thus(9c)From (9a,c) we find that d /ds is a vector which is orthogonal to bothd /ds lies along the principal normal,To decide about the sign we take the cross product ofrotation about :and. Thuswith d /ds and take it as a positiveThus(10a)and(10b)E. Serret-Frenet equationsA set of equations known as the Serret-Frenet equations, which are the intrinsicequations of a curve, are the following. Differentiating the equationwith respect to s, we have(11)Equations (6), (10) and (11) are the Serret-Frenet equations, and are collected below:(12a)(12b)(12c)For a plane curve, = 0, so that(13)2. Geometry of Two-Dimensional Surfaces Embedded in E3Before taking up the main subject of surface theory, it is important to clarify thenotations which are to be used in the ensuing development.In an Euclidean E3, a set of rectangular cartesian coordinates (x,y,z) can always beintroduced.

As before, in E3 a general curvilinear coordinate system will be denoted by i (i== 1,2,3). With these curvilinear coordinates, a surface in E3 will be denoted byconstant, where = 1,2,3. The following convention is adopted which maintains theright-handedness of the two remaining current coordinates: On the surface= constant,the current coordinates are,, where ( , , ) are cyclic.A. First fundamental formLet us consider the surfaceis then given by= constant. In this surface an element of length ds( )(14)where the indices and will assume only the two values different fromcalled the first fundamental form of a surface..

Eq. (14) isB. Unit normal vectorThe unit normal to the surface= constant is defined as(15)where again ( , , ) are cyclic.C. Second fundamental form()A plane containing the normal ( ) to the surface at a point P cuts the surface indifferent curves when rotated about the normal as an axis.

Each curve so generated belongsboth the surface and to the space E3. A study of curvature properties of these curves revealsthe curvature properties of the surfaces in which they lie. We decompose the curvaturevectorgat P of C, defined in Eq. (5), into a vectornnormal to the surface and a vectortangential to the surface as shown below:Thus(16)The vectornis the normal curvature vector at the point P, and is given by(17)whereis its magnitude.

To find an expression forwe consider the equationand differentiate it with respect to s (the arc length along the curve C) to have(18a)Also, differentiating the equationwith respect to, we get(18b)Further,(18c)Thus using Eq. (18b) and (18c) in (18a), we get(19)where(20)The two extreme values ofgiven byare called the principal curvatures kI and kII and their sum is(21)The form(22)is called the second fundamental form.3. Christoffel SymbolsCertain 3-index symbols, known as the Christoffel symbols, show up in a natural waywhen vectors or tensors are differentiated with respect to general coordinates introduced in aspace.