Mathematical Preliminaries Introductory Course on Multiphysics Modelling TOMAZ G. ZIELIŃKI bluebox.ippt.pan.pl/ tzielins/ Table of Contents Vectors, tensors, and index notation. Generalization of the concept of vector........... ummation convention and index notation.........3 Kronecker delta and permutation symbol......... 3.4 Tensors and their representations............. 4.5 Multiplication of vectors and tensors........... 5.6 Vertical-bar convention and Nabla operator....... 6 Integral theorems 7. General idea........................ 7. tokes theorem...................... 8.3 Gauss-Ostrogradsky theorem............... 9 3 Time-harmonic approach 9 3. Types of dynamic problems................ 9 3. Complex-valued notation................. 0 3.3 A practical example.................... 0 Vectors, tensors, and index notation. Generalization of the concept of vector A vector is a quantity that possesses both a magnitude and a direction and obeys certain laws of vector algebra: the vector addition and the commutative and associative laws, the associative and distributive laws for the multiplication with scalars. The vectors are suited to describe physical phenomena, since they are independent of any system of reference.
Mathematical Preliminaries ICMM lecture The concept of a vector that is independent of any coordinate system can be generalised to higher-order quantities, which are called tensors. Consequently, vectors and scalars can be treated as lower-rank tensors. calars have a magnitude but no direction. They are tensors of order 0. Example: the mass density. Vectors are characterised by their magnitude and direction. They are tensors of order. Example: the velocity vector. Tensors of second order are quantities which multiplied by a vector give as the result another vector. Example: the stress tensor. Higher-order tensors are often encountered in constitutive relations between secondorder tensor quantities. Example: the fourth-order elasticity tensor.. ummation convention and index notation Einstein s summation convention A summation is carried out over repeated indices in an expression and the summation symbol is skipped. Example. a i b i A ii A ij b j T ij ij 3 a i b i = a b + a b + a 3 b 3 i= 3 A ii = A + A + A 33 i= 3 A ij b j = A i b + A i b + A i3 b 3 i =,, 3 [3 expressions] j= 3 i= j= 3 T ij ij = T + T + T 3 3 + T + T + T 3 3 + T 3 3 + T 3 3 + T 33 33 The principles of index notation: An index cannot appear more than twice in one term! If necessary, the standard summation symbol must be used. A repeated index is called a bound or dummy index.
ICMM lecture Mathematical Preliminaries 3 Example. A ii, C ijkl kl, A ij b i c j Correct A ij b j c j Wrong! A ij b j c j Correct A term with an index repeated more than two times is correct if: j the summation sign is used, e.g.: i a i b i c i = a b c + a b c + a 3 b 3 c 3, or the dummy index is underlined, e.g.: a i b i c i = a b c or a b c or a 3 b 3 c 3. If an index appears once, it is called a free index. The number of free indices determines the order of a tensor. Example 3. A ii, a i b i, T ij ij scalars no free indices A ij b j a vector one free index: i C ijkl kl a second-order tensor two free indices: i, j The denomination of dummy index in a term is arbitrary, since it vanishes after summation, namely: a i b i a j b j a k b k, etc. Example 4. a i b i = a b + a b + a 3 b 3 = a j b j A ii A jj, T ij ij T kl kl, T ij + C ijkl kl T ij + C ijmn mn.3 Kronecker delta and permutation symbol Definition Kronecker delta. δ ij = { for i = j 0 for i j The Kronecker delta can be used to substitute one index by another, for example: a i δ ij = a δ j + a δ j + a 3 δ 3j = a j, i.e., here i j. When Cartesian coordinates are used with orthonormal base vectors e, e, e 3 the Kronecker delta δ ij is the matrix representation of the unity tensor I = e e + e e + e 3 e 3 = δ ij e i e j. A I = A ij δ ij = A ii which is the trace of the matrix tensor A.
4 Mathematical Preliminaries ICMM lecture Definition Permutation symbol. for even permutations: 3, 3, 3 ɛ ijk = for odd permutations: 3, 3, 3 0 if an index is repeated The permutation symbol or tensor is widely used in index notation to express the vector or cross product of two vectors: e e e 3 c = a b = a a a 3 b b b 3 c i = ɛ ijk a j b k c = a b 3 a 3 b c = a 3 b a b 3 c 3 = a b a b.4 Tensors and their representations Informal definition of tensor A tensor is a generalized linear quantity that can be expressed as a multidimensional array relative to a choice of basis of the particular space on which it is defined. Therefore: a tensor is independent of any chosen frame of reference, its representation behaves in a specific way under coordinate transformations. Cartesian system of reference Let E 3 be the three-dimensional Euclidean space with a Cartesian coordinate system with three orthonormal base vectors e, e, e 3, so that e i e j = δ ij i, j =,, 3. A second-order tensor T E 3 E 3 is defined by T := T ij e i e j = T e e + T e e + T 3 e e 3 + T e e + T e e + T 3 e e 3 + T 3 e 3 e + T 3 e 3 e + T 33 e 3 e 3 where denotes the tensorial or dyadic product, and T ij is the matrix representation of T in the given frame of reference defined by the base vectors e, e, e 3.
ICMM lecture Mathematical Preliminaries 5 The second-order tensor T E 3 E 3 can be viewed as a linear transformation from E 3 onto E 3, meaning that it transforms every vector v E 3 into another vector from E 3 as follows {}}{ T v = T ij e i e j v k e k = T ij v k e j e k e i = T ij v k δ jk e i = T ij v j e i = w i e i = w E 3 where w i = T ij v j δ jk A tensor of order n is defined by nt := T ijk... }{{} n indices e i e j e k..., }{{} n terms where T ijk... is its n-dimensional array representation in the given frame of reference. Example 5. Let C E 3 E 3 E 3 E 3 and E 3 E 3. The fourth-order tensor C describes a linear transformation in E 3 E 3 : C = C : = C ijkl e i e j e k e l : mn e m e n = C ijkl mn e k e m e l e n e i e j = C ijkl mn δ km δ ln e i e j = C ijkl kl e i e j = T ij e i e j = T E 3 E 3 where T ij = C ijkl kl.5 Multiplication of vectors and tensors Example 6. Let: s be a scalar a zero-order tensor, v, w be vectors first-order tensors, R,, T be second-order tensors, D be a third-order tensor, and C be a fourthorder tensor. The order of tensors is shown explicitly in the expressions below. s = v 0 w = v w = v w v i w i = s v = T w = T w T ij w j = v i R = T = T T ij jk = R ik 0 s = T = T : T ij ij = s T = 4 C = 4 C : C ijkl kl = T ij T = v D 3 = v 3D v k D kij = T ij Remark: Notice a vital difference between the two dot-operators and. To avoid ambiguity, usually, the operators : and are not used, and the dot-operator has the meaning of the full dot-product, so that: C ijkl kl C, T ij ij T, and T ij jk T.
6 Mathematical Preliminaries ICMM lecture.6 Vertical-bar convention and Nabla operator Vertical-bar convention The vertical-bar or comma convention is used to facilitate the denomination of partial derivatives with respect to the Cartesian position vectors x x i, for example, u x u i x j =: u i j Definition 3 Nabla-operator.. i e i =. e +. e +. 3 e 3 The gradient, divergence, curl rotation, and Laplacian operations can be written using the Nabla-operator: v = grad s s v i = s i T = grad v v T ij = v i j s = div v v s = v i i v = div T T v i = T ji j w = curl v v w i = ɛ ijk v k j lapl.... ii ome vector calculus identities: s = 0 curl grad = 0 Proof: for i = : s 3 s 3 = 0 s = ɛ ijk s k j = ɛ ijk s kj = for i = : s 3 s 3 = 0 for i = 3: s s = 0 v = 0 div curl = 0 Proof: QED v = ɛ ijk v k j i = ɛ ijk v k ji = v 3 v 3 + v 3 v 3 + v 3 v 3 = 0 QED
ICMM lecture Mathematical Preliminaries 7 s = s Proof: div grad = lapl s = s i = s ii = s + s + s 33 s v = v v curl curl = grad div lapl QED Proof: v ɛ mni ɛijk v k j n = ɛ mniɛ ijk v k jn for m = : ɛ ni ɛ ijk v k jn = ɛ 3 ɛ3 v + ɛ 3 v + ɛ3 ɛ3 v 3 3 + ɛ 3 v 33 = v + v 3 3 v + v 33 = v + v + v 3 3 v + v + v 33 = v i i v ii = v v for m = : ɛ ni ɛ ijk v k jn = v i i v ii = v v for m = 3: ɛ 3ni ɛ ijk v k jn = v i i 3 v 3 ii = v 3 v 3 QED Integral theorems. General idea Integral theorems of vector calculus, namely: the classical Kelvin-tokes theorem the curl theorem, Green s theorem, Gauss theorem the Gauss-Ostrogradsky divergence theorem, are special cases of the general tokes theorem, which generalizes the fundamental theorem of calculus. Fundamental theorem of calculus relates scalar integral to boundary points: b a f x dx = fb fa tokes s curl theorem relates surface integrals to line integrals. Applications: for example, conservative forces.
8 Mathematical Preliminaries ICMM lecture Green s theorem is a two-dimensional special case of the tokes theorem. Gauss divergence theorem relates volume integrals to surface integrals. Applications: analysis of flux, pressure.. tokes theorem Theorem tokes curl theorem. Let C be a simple closed curve spanned by a surface with unit normal n. Then, for a continuously differentiable vector field f: n f n }{{ d } = f dr d C C Formal requirements: the surface must be open, orientable and piecewise smooth with a correspondingly orientated, simple, piecewise and smooth boundary curve C. tokes theorem implies that the flux of f through a surface depends only on the boundary C of and is therefore independent of the surface s shape see Figure. FIGURE Green s theorem in the plane may be viewed as a special case of tokes theorem with f = [ ux, y, vx, y, 0 ] : v x u dx dy = y C u dx + v dy
ICMM lecture Mathematical Preliminaries 9.3 Gauss-Ostrogradsky theorem Theorem Gauss divergence theorem. Let the region V be bounded by a simple surface with unit outward normal n see Figure. Then, for a continuously differentiable vector field f: f dv = f } n {{ d } ; in particular f dv = f n d. d V V n d f V FIGURE The divergence theorem is a result that relates the flow that is, flux of a vector field through a surface to the behavior of the vector field inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. 3 Time-harmonic approach 3. Types of dynamic problems Dynamic problems. In dynamic problems, the field variables depend upon position x and time t, for example, u = ux, t. eparation of variables. In many cases, the governing PDEs can be solved by expressing u as a product of functions that each depend only on one of the independent variables: ux, t = ûx ǔt. teady state. A system is in steady state if its recently observed behaviour will continue into the future. An opposite situation is called the transient state which is often a start-up in many steady state systems. An important case of steady state is the time-harmonic behaviour.
0 Mathematical Preliminaries ICMM lecture Time-harmonic solution. If the time-dependent function ǔt is a time-harmonic function with the frequency f, the solution can be written as ux, t = ûx cos ω t + αx where: ω = π f is called the angular or circular frequency, αx is the phase-angle shift, and ûx can be interpreted as a spatial amplitude.here: ω the angular frequency, αx the phase-angle shift, ûx the spatial amplitude. 3. Complex-valued notation A complex-valued notation for time-harmonic problems A convenient way to handle time-harmonic problems is in the complex notation with the real part or, alternatively, the imaginary part as a physically meaningful solution: exp[iω t+α] ux, t = ûx cos ω t + αx = û Re { {}}{ cosω t + α + i sinω t + α } = û Re { exp[iω t + α] } = Re { û expi α expi ω t } }{{} = Re { ũ expi ω t } ũ where the so-called complex amplitude or phasor is introduced: ũ = ũx = ûx exp i αx = ûx cos αx + i sin αx 3.3 A practical example Consider a linear dynamic system characterized by the matrices of stiffness K, damping C, and mass M: K qt + C qt + M qt = Qt where Qt is the dynamic excitation a time-varying force and qt is the system s response displacement. Let the driving force Qt be harmonic with the angular frequency ω and the realvalued amplitude ˆQ: Qt = ˆQ cosω t = ˆQ Re { cosω t + i sinω t } = Re { ˆQ expi ω t } ince the system is linear the response qt will be also harmonic and with the same angular frequency but in general shifted by the phase angle α: qt = ˆq cosω t + α = ˆq Re { cosω t + α + i sinω t + α } = ˆq Re { exp[iω t + α] } = Re { ˆq expi α expi ω t } }{{} q = Re { q expi ω t }
ICMM lecture Mathematical Preliminaries Here, ˆq and q are the real and complex amplitudes, respectively. The real amplitude ˆq and the phase angle α are unknowns; thus, unknown is the complex amplitude q = ˆq cos α + i sin α. Now, one can substitute into the system s equation Qt ˆQ expi ω t, qt q expi ω t, so that qt = q i ω expi ω t, qt = q ω expi ω t to obtain the following algebraic equation for the unknown complex amplitude q: [ K + i ω C ω M ] q = ˆQ For the Rayleigh damping model, where C = β K K + β M M β K and β M are real-valued constants, this equation can be presented as follows: [ K ω M] q = ˆQ, where K = K + i ω βk, M = M + β M i ω are complex matrices. Having computed the complex amplitude q for the given frequency ω, one can finally find the time-harmonic response as the real part of the complex solution: qt = Re { q expi ω t } = ˆq cosω t + α, where { ˆq = q α = arg q Here, q = Re{ q} + Im{ q} is the absolute value or modulus of the complex number q, and arg q = arctan is called the argument or angle of q. Im{ q} Re{ q}