A Norms
A.l Vector Norms
Definition A.l. Vector norm: A vector norm on mn is a function II II : mn ---+ lR, which fulfills the following properties: (i) llxll 2: 0 for all x E lRn (ii) llxll = 0 iff x = 0 (iii) llx + Yll :::; llxll + IIYII (iv) llaxll = lalllxll
We use the Holder or p-norms, which are defined by
(A.l)
Therefore, we compute, e.g., the 1-norm, 2-norm, and oo-norm as follows
(A.2) i=l
n ) 1/2 llxll2 = ( ~ lxil2 (A.3)
llxlloo = maxilxil· (A.4) The p-norms have the following useful properties
• lxTyl:::; llxiiPIIYIIq, ~ + ~ = 1 (Holder inequality) • lxTyl:::; llxlb IIYII2 (Cauchy-Schwarz inequality) • llxll2 :::; llxllt :::; vlnllxll2 270 A Norms
• llxlloo :S llxll2 :S vfnllxlloo • llxlloo :S llxll1 :S nllxlloo With the help of norms we can define a distance on a vector space, and furthermore, we call a vector space with a norm a normed space.
A.2 Matrix Norms
Definition A.2. Matrix norm: A matrix norm on 1Rnxm is a function II II : 1Rnxm -+ 1R, which fulfills the following properties:
(i) IIAII ~ 0 for all A E 1Rnxm (ii) IIAII = 0 iff A= 0 (iii) IlA + Bll :S IIAII + IIBII for all A, BE 1Rnxm (iv) llaAII = laiiiAII for all a E 1R and A E 1Rnxm
The matrix norm associated to the vector p-norm is defined by the operator norm (A.5)
Other matrix norms are
Frobenius or F-norm (A.6)
n column sum norm (A.7) i=l m IIAIIoo = maxi L laij I row sum norm . (A.8) j=l B Scalar and Vector Fields
Definition B. I. (Scalar field) If we assign to each point in ffi3 defined by the vector r a scalar quantity V (r) (e.g., electric potential, temperature, acous• tic velocity potential}, then V is called a scalar field.
For the illustration of scalar fields we use equipotential lines in the 2D case and equipotential surfaces in the 3D case, where the scalar quantity V(r) is constant (Fig. B.l).
v>
Fig. B.l. Illustration of a scalar field V with the help of equipotential surfaces
Definition B.2. (Vector field) If we assign to each point ffi3 defined by the vector r a vector quantity F(r) (e.g., electric field, magnetic field, mechanical deformation}, then F is called a vector field. 272 B Scalar and Vector Fields
Vector fields are divided into irrotational vector fields (e.g., electrostatic field) and solenoidal vector fields (e.g., magnetic field) as shown in Fig. B.2 (see also Sects. B.12 and B.13).
y y
a) b)
Fig. B.2. (a) Solenoidal vector field; (b) Irrotational vector field
Fig. B.3. Lines of force for the vector field F(r)
The lines of force (see Fig. B.3) are defined by
F(r) x dr = 0, (B.l) which means that in each point of the lines the tangential vector is parallel to the field vector F. In the following, we try to compute the lines of force for the vector field r F(r) = 3 (B.2) r with the help of (B.l). Since we are just interested in the direction, we have to solve rxdr=O. (B.3) By using a Cartesian coordinate system, we obtain B Scalar and Vector Fields 273
r = xex + yey + zez (B.4) dr = dxex + dyey + dzez , (B.5) (B.6) and ex ey ez (ydz- zdy) rx dr= x y z = zdx-xdz (B.7) dx dy dz xdy- ydx Therefore, we can formulate the following three relations
ydz = zdy (B.8) zdx = xdz (B.9) xdy = ydx. (B.lO)
We now search for the line of force including point P0 (x0 , y0 , z0 ). Integration of (B.8) results in y z ln-=ln- (B.ll) Yo zo and zoy = YoZ. (B.l2) Analogously, we can compute the solutions of the other two differential equa• tions
ZoX = XoZ (B.13) YoX = xoy · (B.14) From (B.13) a plane through point Po containing they-axis, and from (B.14) a plane through point Po containing the z-axis is defined. The intersection of the two planes leads to a straight line through the origin, and therefore we obtain the vector field drawn in Fig. B.4, which corresponds, e.g., to the vector field of an electric charge.
Fig. B.4. Lines of force of the vector field F(r) = r/r3 274 B Scalar and Vector Fields B.l The Nabla (V) Operator
First, we recall that a scalar function may depend on one or more variables, e.g., using Cartesian coordinates, a function can be denoted by
f = f(x,y,z).
The partial derivatives read as
of of of ox' 8y' 8z.
The nabla operator V is defined in Cartesian coordinates by
(B.15)
where ex, ey and ez are the unit vectors in x-, y-, and z-directions. The interaction between the nabla operator and a scalar or a vector field yields its geometric significance.
B.2 Definition of Gradient, Divergence, and Curl
We introduce a scalar function V with nonzero first-order partial derivatives with respect to the coordinates x, y, and z, and a vector field F with compo• nents Fx, Fy, and Fz. Then, the following operations are defined:
1. Gradient of a scalar:
As can be seen, the result of this operation is a vector.
2. Divergence of a vector:
div F = V . F = 8Fx + 8Fy + 8Fz . ax 8y 8z Therefore, the result of this operation is a scalar value. B.3 The Gradient 275
3. Curl of a vector:
!!E.._8y 8Fy)8z _ ( 8Fm _!!E.. curl F = \7 X F = a;ax 8/oy a;az - 8z 8x 8Fy _!!E.. Fx Fy Fz 8x 8y The result of taking the curl of a vector is again a vector.
B.3 The Gradient
We will consider the scalar function V(x,y,z) with its partial derivatives 8Vf8x, 8Vj8y, 8Vf8z and dependent on a point P = (x, y, z). In a first step we calculate the total differential of V av av av dV =ox dx+ oy dy+ oz dz. (B.16)
Now, we define a point P' infinitely close toP by P' = (x+ dx, y+ dy, z+dz). By calculating the vector dP = P' - P, which has the components dP ( dx, dy, dz)T, we can write (B.l6) as
(B.17)
(B.l8)
For the geometrical illustration of the gradient, consider an equipotential sur• face, i.e., a surface with V = canst. (see Fig. B.5). Hence, for all differential
vv
dP
p
Fig. B.5. The gradient is orthogonal to a constant potential surface
displacements from P to P' on this surface dV = 0 holds, and therefore,
VV · dP = 0. (B.19) 276 B Scalar and Vector Fields
From the definition of the scalar product it is clear that VV and dP are orthogonal. In this situation the displacement from P to P' points into the direction of increasing V, as shown in Fig. B.6, and the scalar product VV · dP is positive.
vV
p
Fig. B.6. Geometrical representation of the gradient
From the foregoing arguments, we conclude that VV is a vector, perpen• dicular to the surface on which V is constant and that it points in the direction of increasing V. As an example we consider a function r(x, y, z), which defines the distance of a point P from the origin (0, 0, 0). The surface r = con st. is a sphere of radius r with center (0, 0, 0), whose equation is given by
r = J x2 + y2 + z2 .
Therefore, the gradient calculates as
8r X X ax Jx2 + y2 + z2 r 8r y 8y r 8r z 8z r Vr = xex + yey + zez r r r Geometrically speaking, V r points in the direction of increasing r, or towards spheres with radii larger than r. B.4 The Flux 277 B.4 The Flux
Definition B.3. (Flux) The vector field F(r) and a corresponding surfacer as shown in Fig. B. 7 are given. The vector n denotes the normal unit vector of the differential surface di'. Therefore, the differential flux d'ljJ through di' is defined by d'I/J=F·dr=F·ndr. (B.20) The total flux 'ljJ computes as
'1/J= IF· dr. (B.21) r
In the following, we want to compute the flux 'ljJ of the vector field F(r) = r
Fig. B.7. Flux through the surfacer through the square r with side length h according to Fig. B.8. With the normal unit vector n = ex and dr = dy dzex we obtain
h h 'ljJ = I I (-hex + yey + zez) ·ex dy dz 0 0 h h = -h 11 dydz 0 0 = -h3. (B.22)
The total flux 'ljJ through a closed surface S is given by 278 B Scalar and Vector Fields
"'
z y t Jf h h
-h
Fig. B.S. Flux 'lj; through the square with area h2
(B.23) and defines whether we have sources ('1/J > 0) or sinks ('1/J < 0) within r. A very important property of the flux 'ljJ defined by a closed surface is given by (see Fig. B.9)
f F-dr+ f F-dr= f F-dr. (B.24) r1uro r2uro nur2
Fig. B.9. Flux through the closed surface HUH
B.5 Divergence
Definition B.4. (Divergence) The vector field F(r) is given. If we divide the flux 'ljJ, defined by a closed surface r, by the corresponding volume Jl and let the volume Jl tend to zero, then the obtained value is called the divergence (source density) B.5 Divergence 279
divF = N!!:o ~iF· dr = 1~ lr. (B.25)
Let us now consider the closed surface of a differential cube (see Fig. B.lO) and the general vector field F(r) = Fxex + Fyey + Fzez. In a first step, let us
(x-dx/2,y,z) (x+dx/2,y,z)
Fig. B.lO. Flux through a cube compute the differential flux through the hatched surfaces F · dr = [F(x + dx/2, y, z)- F(x- dx/2, y, z)]· ex dy dz oFx dx ( oFx dx)] ~ [ Fx(x,y,z)+ 0x 2 - Fx(x,y,z)-ox2 dydz oFx = ox dxdydz. (B.26) Analogously, we obtain the contribution of the other two directions, and thus, the differential flux
d"'' = ( oFx oFy oFz) d d d (B.27) 'f/ OX + oy + oz X y z. Since the differential volume dft is equal to dx dy dz, we end up with the following expression for the divergence of a vector field in Cartesian coordi- nates . F _ oFx oFy oFz d (B.28) IV - OX + Oy + OZ ' or, by using the nabla operator, divF = V ·F. (B.29) 280 B Scalar and Vector Fields B.6 Divergence Theorem (Gauss Theorem)
By the definition of the divergence (see (B.25)) we get d1/J = V · F df? (B.30) 1/' = l V · F df?. (D.31) On the other hand, we have the relation for the flux 1/J according to (B.23). Combining these two expressions for the flux results in
1/J = { V · F df? = j F · dr . (B.32) ln Y"r(n) This equality between the two integrals tells us that the flux of the vector F through the closed surface r is equal to the volume integral of the divergence of F over the volume f.? enclosed by the surface r.
F
Fig. B.ll. Radial vector field
Consider a radial vector field F as shown in Fig. B.ll, and assume that the magnitude ofF is constant in all points on a sphere centered at P. To compute the flux of the vector field F through a spherical shell of radius R, we note that ds and F are colinear and in the same direction 1/1 = t F · dr = F i dr = 4nR 2 F . From the divergence theorem, (the flux is nonzero) we conclude V·F:fO.
B.7 The Circulation
The circulation of a vector field F(r) along a closed contour Cis given by the closed-line integral B.8 The Curl 281
Z = iF ·ds. (B.33) Therefore, the important property follows (see Fig. B.l2)
1 F · dr + 1 F · dr = 1 F · dr. (B.34) ~~u~ ~2u~ ~~u~ If the circulation along a closed curve C is not equal to zero, then we say the
Fig. B.12. Circulation along the closed line C1 U C2 closed line contains eddies.
B.8 The Curl
Definition B.5. (Curl) We consider a point defined by r {Fig. B.13), in which the curl of the vector field F has to be computed. Furthermore, we define a closed line C enclosing the area r and consider the circulation along C. If the area r tends to zero, we obtain the definition of the curl by
_ . fc F · ds _ dZ n · curl F - ).t~0 r - dF . (B.35)
The vector curl F is obtained by a separation in the three directions of the unit vectors ex, ey, and ez
curl F = ( e, · curl F)e, + (ey · curl F)ey + (ez · curl F)ez . (B.36)
The circulation for the differential square in Fig. B.14 is given by
dZ, = ( F(x, y, z- dz/2)- F(x, y, z + dz/2) ) · ey dy + ( F(x,y + dyj2,z)- F(x,y- dyj2,z)) · ez dz
~ (aFz _8Fy) dydz. (B.37) 8y az 282 B Scalar and Vector Fields
0 Fig. B.13. Curl in a point defined by r
Therefore, we obtain the x-component of curl F with dT = dy dz 8Fz 8Fy ex · curl F = {)y - 0 z . (B.38)
Analogously, the y- and z-component of curl F can be computed, and the full
Fig. B.14. x-component of curl F vector in Cartesian coordinates reads as
(B.39) B.9 Stoke's Theorem 283 or with the help of the nabla operator
a a a curl F = V x F = Bx By Bz (B.40) Fx Fy Fz
B.9 Stoke's Theorem
We consider the vector field F on the surface r with fixed oriented contour C as shown in Fig. B.l5. For a differential surface dTv, we obtain according
Fig. B.15. Vector field F on the surface r with fixed oriented contour C to (B.35)
dZv = n(rv) · curlF(r)dTv Zv = JcurlF(r) · drv, (B.41) rv and Z = l curl F · dr . (B.42) Furthermore, according to the definition of the circulation Z (see B.33), we get the following relation
Z = F · dr = curl F . dr . (B.43) ic r J 284 B Scalar and Vector Fields
For a radial vector field F as shown in Fig. B.ll, the closed-line integral along a circle C of constant radius
£F· ds is zero, and therefore, the curl of this vector field V x F is zero, too.
B.lO Green's Integral Theorems
The integral theorems of Green can be derived from the divergence theorem. For this purpose, we first introduce the Laplace operator by
[)2 [)2 [)2 L1 = V . V = fJx2 + [)y2 + [)z2 . (B.44)
This differential operator can be applied to scalar as well as vector quantities
L1 V = div (grad V) (B.45) L1F = (L1Fx)ex + (L1Fy)ey + (L1Fz)ez. (B.46) Setting a vector F equal to V1VV 2 and using the divergence theorem, we obtain according to (B.32)
fn v · (V1 VV2) dD = £(V1 VV2) . dr. (B.47)
Since the term V · (V1VV 2) can be expressed by (see (B.54) below)
v. (V1 VV2) = V1L1V2 + vv1 · vv2, (B.48) we get the following integral theorem, called Green's first integral theorem
fn V1L1V2 dD + fn vv1 · vv2 dD = £V1 ~';: dr. (B.49)
By substituting V1 with V2 and vice versa in (B.47) and subtracting the re• sulting equation from (B.47), we achieve Green's second integral theorem
(B.50)
In addition, Green's first integral theorem in vector form is
J(V X u · V X v - u · V X V X v) d.f2 !.7
= J( U X V X V) · ll dT , (B.51) r B.12 Irrotational Vector Fields 285 and Green's second integral theorem in vector form reads as
I (u · V x V x v- v · V x V xu) dD (}
= I (v X v X u - u X v X v) . n dr. (B.52) r
B.ll Application of the Operators
By using the definitions of gradient, divergence, and curl in Cartesian coordi• nates, the following relations hold:
V(V1 V2) = v1 vv2 + v2 vv1 (B.53) V · (VF) = VV · F + F · VV (B.54) v. (Fl X F2) = F2. v X Fl- Fl. v X F2 (B.55) v X (VF) = vv X F - F X vv (B.56) ..::lF = v (V . F) - v X (V X F) . (B.57)
These relations combine the essential differential operators and build up a basis for the description of physical fields.
B.12 Irrotational Vector Fields
We consider a vector field F, which is given as the gradient of a scalar potential F = VV. The computation of a line integral from point A to point B yields
B I (VV) · dr = V(B) - V(A). (B.58) A Therefore, for any closed contour within this vector field, the following relation holds f (V V) · dr = 0. (B.59) This result proves that any vector field that can be expressed by the gradient of a scalar potential is irrotational. Furthermore, the local quantity, given by the curl of the vector field, is zero
v X VV=O. (B.60) 286 B Scalar and Vector Fields
Fig. B.16. Domain for solenoidal vector field
B.13 Solenoidal Vector Fields
We will consider the solenoidal vector field V x F for a domain as displayed in Fig. B.16. This domain shall consist of two subdomains defined by their surfaces T1 and r 2 with their related contours C1 and C2 . By using Stoke's theorem, we obtain the following relation
1 (V x F) . dr = 1 (V x F) . n1 dr + 1 (V x F) . n2 dr Jr Jr1 Jr2 =1 F·dr+1 F·dr Jc1 Jc2 = 0. (B.61)
Thus, the total flux (global quantity) is zero, and, furthermore, the local solenoidality, too
v 0 (V X F)= 0. (B.62) c Appropriate Function Spaces
Let us define the derivative of order a with respect to the multi-index a, with Ia I = Li ai and ai E IN, as follows alalv D "v := --,-a-a-=-,----=-a-n . (C.1) x 1 · · · Xn For example, the partial derivatives of order 2 in lR? can be written as D"v with a= (2, 0), a= (1, 1) or a= (0, 2), since lal = a1 + a2 = 2 is fulfilled for all three cases
a= (2, 0)
a = (1, 1)
a= (0, 2)
Definition C.l. Continuously differentiable functions: Let f? be a closed domain in IR n and let C (D) denote the space of continuous functions on f?. Now, the space of up to order m continuously differentiable functions is given by cm(f?) = {v: f?---+ IR I D"v E C(D), lal::::; m}. (C.2) If the function v is infinitely often continuously differentiable on f?, we write v E c=(n). For the function u(x) shown Fig. C.1 the following inclusions hold (with v(x) = u'(x)) 288 C Appropriate Function Spaces
L--+---+---x 2 2 2 Fig. C.l. Example of a C 1 function
Definition C.2. Square integrable functions: Let n be a closed domain in IRn. Then, the function u is called square integrable, if it fulfills the following relation Jlu(x)l 2 dx < oo. (C.3) Q We denote 2 L2(n) = {u: n-+ IR If I u(x) 1 dx < oo}. (C.4) Q
For example, the function f(t) with the definition
1 for 0 < x < 2 f (t) = { 0 for x = 0 (C.5) -1 for -2 :S x < 0 belongs to the space £ 2 ( -2, 2) (see Fig. C.2).
u(x)
~------~------+---X -1 0
-----+-1
Fig. C.2. Function u(x) = sgn(x) in the interval (-2,2)
Analogously to the above definition, we obtain the definition for Lp(D)• spaces for p E [1, oo ). C Appropriate Function Spaces 289
Definition C.3. Lp(D)-spaces: Let D be a closed domain in lRn. Then, the space of p-integrable functions is given by
Lp(D) = {u: n--+ lRI I iu(x)IPdx < oo}. (C.6) n
Let us assume that the function u has a continuous derivative u'. According to the formula for partial integration, we have for each continuously differentiable function cp with cp( a) = cp(b) = 0 the following relation
b b I u(x)cp'(x) dx =-I u'(x)cp(x) dx. (C.7) a a
With the help of (C. 7), we can define the derivative of functions, which have no finite derivative in the classical sense. If u and w denote integrable functions that fulfill the following relation
b b I u(x)cp'(x) dx =-I w(x)cp(x) dx (C.8) a a for all differentiable functions cp with cp( a) = cp(b) = 0, then the function w is called the derivative of u in the weak sense (with respect to x). The function
-1 0 Fig. C.3. Example of a function in H 1 (a,b)
u defined by (see Fig. C.3)
u(x) = { x + 1 for -1 ~ x ~ 0 1 - x for 0 < x ~ 1 will have no derivative in the classical sense at x = 0. Applying partial inte• gration for differentiable functions cp(x) with cp( -1) = cp(1) = 0, we obtain 290 C Appropriate Function Spaces
1 0 1 Ju(x) 0 =-J ~ -[J ~(x) dx + l( -l)l'(x) dx] + 1'(0) :" ~(0) . Therefore, in the weak sense of differentiation we obtain u' ( x) = { 1 for -1 ::; x < 0 -1 for 0 < x ::; 1 with an arbitrary value for u'(O). Definition C.4. Sobolev space: Let f? be a domain in lR n. The functional space (C.9) is called Sobolev space W;'(f?). The partial derivatives ofu are defined in the weak sense. The appropriate norms on Sobolev spaces are defined by (C.lO) If we restrict p to two, then we obtain a Hilbert space (W2(f?) = Hm(f?)) with the scalar product (u,v) = J( L DauDav) dx. (C.ll) n lal<:::m For example, the function u(x) is in the space H 1 (a, b), if u'(x) exists and is within the space L2 (a, b). The norm is computed via b b iiuiiH'(a,b) = J(u(x))2 dx + J(u'(x))2 dx, (C.12) a a C Appropriate Function Spaces 291 and the scalar product as follows b b (u,v)H'(a,b) = Ju(x)v(x) dx + Ju'(x)v'(x) dx. (C.13) a a Definition C.5. Let fl be a domain in IRn and denote by C0 (fl) the space of infinitely often differentiable functions with zero boundary values. Then we write for the closure of Cif ( fl) with respect to the H 1 norm (C.l4) Definition C.6. Partial Integration: Let fl C IRn, n = 2, 3 be a domain with smooth boundary r. Then, for any u, v E H 1 ( fl) the following relation holds au v dx = r uv n. e; ds _ u~ dx. (C.15) f oxi } r 1 oxi n n In (C.15) n denotes the outer normal and {l the considered domain fl with boundary r. By a multiple application of (C.15), we arrive at Green's formula j L1uvdx= ~~~vds- jcvu)T\lvdx (C.l6) n r n D Solution of Nonlinear Equations In this section we are concerned with the solution of systems of nonlinear equations. As an example, we will consider the nonlinear Poisson equation, given as follows -V · c(\Vu\)Vu + f = 0 (D.l) u = 0 on r. (D.2) This defines a nonlinear operator :F that allows us to rewrite (D.l) and (D.2) as :F(u) = 0. (D.3) The weak formulation of (D.l) and (D.2) for all test functions v E HJ reads as Jc(JVu\)Vv·VudD- JvfdD=O. (D.4) n n By applying the finite element method, we arrive at the following algebraic system K(y)y = i_, (D.5) with the matrix K E IRnxn, f E IRn, y E IRn and n the number of unknowns. Since we cannot solve (D.5) explicitly, we have to establish an approximate solution by setting up a series Yk (k = 0, 1, 2, 3, .. ) that is supposed to converge to the correct solution. Concerning the rate of convergence, we will restrict the discussion to the following types: Definition D.l. Convergence: Let y* E IRn be the exact solution. Then • Yk converges towards y* q-quadratically (q stands for quotient), if there exists a C such that (D.6) 294 D Solution of Nonlinear Equations • '!lk converges towards 'Q* q-linearly with the q-factor O" E (0, 1), if (D.7) In general, a q-quadratically convergent algorithm is preferable to a q• lincarly convergent one. However, we always have to take into account thP numerical cost for one iteration. Therefore, in some cases the method with the slower convergence rate can even be faster. Since we solve (D.5) numerically by computing a series of approximating solutions '!lk, the question of the stopping criterion is of great importance. In general we distinguish between the following two types of stopping criteria: 1) Error criterion: We take the solutions of two successive iteration steps and define an ab• solute accuracy cabs by (D.8) and a relative accuracy ere! by (D.9) which has to be achieved. However, in some analysis the true solution may still be far away, although the above-defined stopping criteria are fulfilled. This may particularly occur in the solution methods, that have to use a line search (see Sect. D.l) to avoid possible divergence during early steps of the iteration process or due to nonmonotonic material relations. Then, it can happen that the control parameter becomes very small, which re• sults in almost no difference between '!lk+l and '!lk· 2) Residual criterion: By computing the residual of the obtained solution, we can define an ab• solute accuracy c~~s" by (D.lO) as well as a relative accuracy c~~~ by (D.ll) As shown in Fig. D.l, according to the problem type, this stopping criterion may also be reached too early. As a consequence of the above discussion, it is preferable to check both stopping criteria. D.l Fixed-point Iteration 295 abs s_ ------ !! Fig. D.l. Obtained solution uk+ 1 is still far away from the true solution u* D.l Fixed-point Iteration The simplest method of solving (D.5) is to rewrite it as a fixed-point equation (D.12) This will result in the following sequence lik+l = K-1(llk)[. (D.13) K(:!!k)llk+l = [. · (D.14) Thus, we can write the damped fixed-point iteration method as follows K(:!!k)Ll:!! = [.- K(:!!k)llk = r.(llk) (D.15) lik+ 1 = lik + 'f/d:!!. (D.16) The nodal vector r.(:!!) is known as the residual of the problem and a solution is given by the set of nodal values :!!, for which the residual is zero. The scalar parameter rJ E [0, 1] is introduced to control the possible divergence during the early steps of the iteration process or due to nonmonotonic material relations. A common algorithm to compute rJ is a line search (see [144]) defined by IG(ry)l -+min, (D.17) with G(ry) = Ll:!! · r.(llk + rJd:!!). (D.18) One simple method of approximating the optimal 'fJ is as follows 296 D Solution of Nonlinear Equations 1. Evaluate 91 = G(O.l) and 92 = G(l.O) 2. Calculate the straight line l(91,92) between 91 and 92 3. Calculate the value 'fJ = 1 ~~~~-=-~~) for which l(91, 92) = 0 holds A graphical interpretation of the fixed-point method is given in Fig. D.2. K(y)y Fig. D.2. Graphical interpretation for solving a nonlinear equation using the fixed• point method D.2 Newton's Method Let us introduce the following linearization of the nonlinear operator :F(u) at (D.l9) with Uk+1 = Uk + s. The term :F'(uk)[s] denotes the Frechet - derivative of the nonlinear operator :F at Uk in the direction of s and is defined as follows Definition D.2. Frechet - derivative: Let X andY be two normed vector spaces and D C X an open domain. The operator :F : D -+ Y is differentiable in the sense of Frechet at x, iff there exists an operator A : X -+ Y, so that for ally ED F(y) = :F(x) + A(y- x) + R(x,y), D.2 Newton's Method 297 with lim IIR(x, Y)ll = 0 y--+x lly-xll is fulfilled. A is the Frechet derivative :F' (x). Therefore, Newton's method reads as F'(uk)[s] = -:F(uk) (D.20) Uk+l = Uk + S. (D.21) Analogously to the fixed-point method, a line-search parameter may accelerate the convergence, and in addition may guarantee a global convergence of the Newton method. A graphical interpretation of Newton's method is displayed in Fig. D.3. K(y)y I I I I I If 1- I I I I I I I I .l! .\!, !6 y, Fig. D.3. Graphical interpretation for solving a nonlinear equation using Newton's method To derive the Frechet derivative :F' and Newton's method for the nonlinear Poisson equation given in (D.l), we first compute the difference between :F(u+ s) and :F(u) in the weak formulation for arbitrary test functions v E HJ Jc(IV(u + s)I)Vv · V(u + s) d.f?- Jc(!Vui)Vv · Vud.f?. (D.22) n n Now, we will add to and at the same time subtract from (D.22) the term J c(IV(u)I)Vv · V(u + s) d.f?, and obtain n 298 D Solution of Nonlinear Equations I (E(!V(u + s)l)- E(IVul)) Vv · V(u + s) dft +I E(!Vu!)Vv · V s dft. n n The term E(IV(u + s)l)- E(!Vul) can be approximated as follows t:(IV'lu + s)j)- t:(jVuj) ~ t:'(iVui) (iV'(H + s)l -IVul) . 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Index absorbing boundary condition, 121 Burger's equation, 118 acoustic crosstalk, 256 circulation, 280 density, 108 CMUT, see micromachined field, 107 coarse grid operator, see grid impedance, 108 coarsening linear wave equation, 113 agglomeration technique, 205 nonlinear wave equation, 115 function, 203 particle velocity, 108 process, 203 pressure, 108 coil quantities, 107 current-loaded, 98 sound intensity, 108 voltage-loaded, 147 sound power, 109 condition number, 192 sound-intensity level, 109 configuration sound-power level, 109 deformed, 38 sound-pressure level (SPL), 109 initial, 38 velocity potential, 114 congruency, 184 actuator, see mechatronic conjugate gradient (PCG) method, see adiabatic, 110 preconditioned agglomeration technique, see coarsening contact mechanics algebraic multigrid, see multigrid condition, 240 Ampere, 63 pressure-displacement relation, 240 approximation tangent stiffness matrix, 241 B-H - curve, 96 continuity equation, 111 auxiliary matrix, 202 convergence, 293 Coulomb-gauge, see gauge B-H - curve, see approximation coupling Biot-Savart's law, 76 electromagnetics-mechanics, 143 boundary condition, 9 electrostatics-mechanics, 133 Dirichlet, 9 mechanics-acoustics, 165 essential, 10 piezoelectrics, 175 natural, 10 coupling mechanisms, 4 Neumann, 9 Crank-Nicolson scheme, 13 bulk viscosity, 115 crosstalk, see acoustic, see mechanical 308 Index curl, 275, 281 electrostatic-mechanical system, 133 calculation scheme, 137 damping, 50 Euler equation, 112 modal, 50 Eulerian coordinate, 38 Rayleigh model, 50 Everett-function, 185 dPn~ity, ~7 derivative Faraday, 64 Frechet, 296 Fay solution, 129 global/local, 32 ferroelectricity, 77, 177 weak sense, 289 ferromagnetic, 73 design process, 3 finite element, 7 CAE-based, 3 assembling procedure, 20 experimental-based, 3 compatible, 18 diamagnetic, 73 conforming, 18 dielectric remnant, 77 edge, 29 differential operator, 37 formulation, 8 diffusion equation, 71 hexahedral, 28 diffusivity of sound, 117 infinite element, 121 displacement current density, 64 isoparametric, 20 divergence, 274, 278 Lagrangian, 16 theorem, 280 method, 7 Nedelec, 29 elasticity modulus, 41 nodal, 16 electric quadrilateral, 23 charge density, 62 tetrahedral, 27 conductivity, 62, 76 triangular, 26 current, 63 finite element/boundary element current density, 62 method, 155 field intensity, 62 fixed-point iteration, 295 flux density, 62 flux, 277 permittivity, 62 force polarization, 62, 177 electromagnetic, 145 scalar potential, 72 electrostatic, 134 specific resistivity, 76 formulation electrodynamic loudspeaker, see strong, 9 loudspeaker variational, 9 electromagnetic weak, 9 energy, 146 Fubini solution, 128 field, 61 functional spaces, 287 force, 145 Lp, 288 interface conditions, 77 continuously differentiable, 287 quasistatic field, 69 Hilbert, 290 electromagnetic-mechanical system, 143 Sobolev, 290 calculation scheme, 152 square integrable, 287 electromotive force, 65 weighted Sobolev, 88 electrostatic energy, 134 Galerkin, 10 field, 72 method, 10 force, 134 semidiscrete formulation, 12 Index 309 gauge, 70 field intensity, 62 Gauss, 67 flux, 64, 99 Gauss theorem, see divergence hard material, 73 geometric multigrid, see muiltigrid hysteresis, 74 gradient, 275 inductance, 99 deformation, 38 induction, 62 displacement, 39 permeability, 62, 72 of a scalar, 274 reluctivity, 73 Green's integral theorem, 284 remnant field, 73 scalar form, 284 scalar potential, 75 vector form, 284 soft material, 73 grid vector potential, 69 coarse, 203 magnetic valve, 143, 239 coarse-grid operator, 207 overexcitation, 246 complexity, 213 premagnetization, 244 fine, 203 switching cycle, 246 magnetization, 62 harmonic distortion, 225 magnetomechanical system, see Helmholtz decomposition, 45, 198 electromagnetic-mechanical Hook's law, 41 system hysteresis, 181 Maxwell's equations, 61 Preisach model, 181 mechanical acceleration, 37 induced electric voltage, 102 axisymmetric stress-strain, 44 inductance, see magnetic contact, 240 infinite finite elements, 121 crosstalk, 255 interpolation function, 18 damping, see damping irrotational, 68, 272 field, 35 vector field, 285 plane strain, 43 plane stress, 43 Jacobi, 33 strain, 39 matrix, 33 stress, 35 Khokhlov-Zabolotskaya-Kuznetsov stress-stiffening effect, 141 (KZK) equation, 118 yield stress, 47 Kuznetsov's equation, 115 mechanical-acoustic system, 165 calculation scheme, 168 Lagrange multiplier, 156 mechatronic, 1 Lagrangian actuator, 1 coordinate, 38 sensor, 1 updated formulation, 144 micro machined Lame - parameters, 41 capacitive ultrasound array (CMUT), line search, 295 253 local support, 17 motional electromotive force, 67, 143 locking effect, 58 method, 156, 158, 222 Lorentz force, 4, 63 moving body loss factor, 50 electric field, 138 loudspeaker, 4, 154, 221 magnetic field, 143 moving coil magnetic current-loaded, 154 310 Index voltage-loaded, 154 ceramics, 177 moving-material method, 157, 159, 223 cofired multilayer, 248 moving-mesh method, 138, 157 direct effect, 1 7 5 multigrid, 191 inverse effect, 175 algebraic, 201 systems, 175 F;f'Omf'tric, 195 Poisson ratio, 41 method, 193 polymers, 177 nested, 199 power transformer, 228 multilayer actuator, see piezoelectric preconditioned conjugate gradient (PCG) method, 191 nabla operator, 274 predictor-corrector algorithm, 14, 15 Navier's equations, 37 Preisach Newmark scheme, 14 function, 182 Newton method, 296 model, 181 electromagnetics, 93 operator, 182 mechanics, 53 prestessing, 141 norms, 269 principle of virtual work, 134, 145, 149 Holder, 269 prolongation, 193 matrix, 270 operator, 193, 206 p-norms, 269 pulse-echo mode, 256 vector, 269 numerical computation Rayleigh damping model, see damping electromagnetics, 82 remnant magnetic field, see magnetic electromagnetics-mechanics, 152 restriction, 193 electrostatics, 81 operator, 193 electrostatics-mechanics, 137 geometric nonlinear case, 51 scalar linear acoustics, 119 acoustic velocity potential, 114 linear elasticity, 4 7 electric potential, 72 mechanics-acoustics, 168 field, 271 nonlinear acoustics, 123 magnetic potential, 75 nonlinear electromagnetics, 93 sensor, see mechatronic piezoelectrics, 178 shape function, see interpolation numerical integration, 31 function Gaussian quadrature, 31 shear modulus, 41 shear viscosity, 115 operator shock-formation distance, 129 complexity, 213 single crystals, 177 nonlinear, 293 skin depth, 71 paramagnetic, 73 effect, 70 parameter of nonlinearity, 115 smoothing partial differential equation, 8 overlapping block-smoothers, 198 hyperbolic, 14 block-Gauss-Seidel, 198 parabolic, 13 Gauss-Seidel backforward, 195 penalty formulation, 87 Gauss-Seidel forward, 195 penetration depth, see skin depth hybrid, 211 permeability, see magnetic permeability operator, 207 piezoelectric, 175 post, 194 Index 311 pre, 194 time discretization, 12 Sobolev space, see functional spaces effective mass formulation (hyper• solenoidal, 68, 272 bolic), 15 vector field, 286 effective mass formulation (parabolic), solid/fluid interface, 166 14 sound velocity, 107 effective stiffness formulation sound-field impedance, 108 (hyperbolic), 16 SPL, see acoustic effective stiffness formulation stack actuator, see piezoelectric (parabolic), 14 state equation, 110 explicit (hyperbolic), 15 Stoke's theorem, 283 explicit (parabolic), 13 stopping criterion, 294 implicit (hyperbolic), 15 error, 294 implicit (parabolic), 13 residual, 294 transducing mechanisms, 3 strain, see mechanical transformer, see power transformer strain tensor trapezoidal integration scheme, 13 Green~Lagrangian, 40 linear, 40 vector field, 271 stress tensor irrotational, 272 1st Piola~Kirchhoff, 52 solenoidal, 272 2nd Piola~Kirchhoff, 52 virtual work, see principle Cauchy, 36 Voigt notation, 37, 40 stress-stiffening effect, 141 surface integration, 34 wave TEAM (Testing Electromagnetic longitudinal, 46, 107 Analysis Methods), 198 number, 108 tensor shear, 46 of dielectric constants, 176 weighted regularization, 88 of elasticity moduli, 42, 176 Westervelt equation, 118 of piezoelectric moduli, 176 wiping-out, 183 test function, 9 thermal strain, 47 yield stress, 4 7