Geometric Variations of Surface Area and Volume

Chapter V: Shape Analysis and Energy Stability of Conductive Liquids

5.4 Geometric Variations of Surface Area and Volume

As a sanity check, we test the second variation ofC on a spherical capacitor occupying the region between two concentric spheres of inner radius r_a and outer radius r_b. The exact solution to the electric field within the capacitor is given by

ψ= r_a

r_a−r_b 1−r_b r

, ψ(ra) = 1, ψ(rb) = 0, (5.97) where r is the radial distance and the outward normal vector n = −g_r on the inner surface. We consider a configuration mapχ(R, )such that the inner radiusr_ain spatial coordinates is mapped fromR_ain material frame,

r_a=χ(Ra, ) =R_a+. (5.98) The rate of spatial deformation vector v = g_r in this case is purely normal to the inner surface. The exact solution to the self-capacitance C[ra] and its first two total derivatives in for any radiusr_a can be easily obtained,

C[r_a] = 4πrar_b r_b−ra

, dC[ra]

d = 4πr_b²

(rb−ra)², d²C[ra]

d² = 8πr²_b

(rb−ra)³. (5.99) The auxiliary problem of the Eulerian derivative ψ⁰ also admits the analytic solution,

ψ⁰ = r_b (rb−r_a)²

r_b

r −1, ψ⁰(ra) =−v·dψ= r_b/ra

r_b−r_a, ψ⁰(rb) = 0. (5.100) We also compute the necessary ingredients needed in the integral formula of the first and the second derivatives,

n·dψ|ra = r_b/r_a r_b−ra

, n·dψ⁰|ra = r²_b/r²_a

(rb−ra)², H= 1 ra

, v_n=−1. (5.101) It can be easily verified that, with quantities from expression (5.101) substituted, the surface integrals (5.49) and (5.92) are indeed evaluated to the identical results from the exact solution (5.99).

First and second derivatives of area differential

Variational calculations of area element is much easier with convective Lagrangian co- ordinates. We start with an equivalent expression of a spatial area element dγ,

dγ =n·ndγ. (5.102)

Substituting Nanson’s relation (5.21) into (5.102) and taking total derivative with re- spect to yield

ddγ

d =n·d(ndγ) d +dn

d ·ndγ

=n·∂JF^−>

∂ NdΓ +1 2d|n|²

d dγ Note |n|= 1

=n·^hJ(divv)F^−>−Jl^>F^−>iNdΓ

=n·^h(divv)−l^>indγ = divv−nl^>ndγ. (5.103) Using the adapted curvilinear coordinates for surface γ, we arrive at a convenient ex- pression for the first total derivative (or deformation rate) of spatial area element,

d(dγ)

d = divγvdγ = divγv_γ−2hvndγ. (5.104) Note from the calculation of (5.103) that, the following identity

dn

ddγ = d(ndγ)

d −nddγ d

= divv−l^>ndγ−n divv−nl^>ndγ =^h nl^>nn−l^>nⁱdγ (5.105) must hold for all area differentials dγ which is only possible if

dn

d = nl^>nn−l^>n. (5.106) Equation (5.106) is the first variation of surface normal vectornas boundaryγ deforms.

We then apply a similar strategy to derive the second total derivative of spatial area element which yields

d²n·ndγ d²

=n·d²(ndγ)

d² +d²n

d² ·ndγ+ 2dn

d ·d(ndγ)

d note d²|n|² d² = 0

=n·∂²JF^−>

∂² NdΓ −dn d ·dn

d dγ+dn

d ·d(ndγ) d +dn

d ·d(ndγ) d

=n·∂²JF^−>

∂² NdΓ +dn

d ·nddγ

d + nl^>nn−l^>n·(divv)−l^>ndγ

=n·∂²JF^−>

∂² NdΓ +nl l^>ndγ−(nl^>n)²dγ, (5.107)

where∂²JF^−>/∂² can be explicitly expanded in aid of (5.28) and (5.30),

∂²JF^−>

∂² = 2∂J

∂

∂F^−>

∂ +∂²J

∂²F^−>+J∂²F^−>

∂²

=−2J(divv)l^>F^−>+J^h(divv)²+ div( ˙v)−tr(l l)ⁱF^−>

+Jl^>l^>F^−>−J(dv˙^>−l^>l^>)F^−>

=−2(divv)l^>+ (divv)²−tr(l l) + 2l^>l^>+ div( ˙v)−dv˙^>JF^−>. (5.108) Applying identity (5.108) to (5.107) finally leads to the spatial form of the second derivative of area differential,

d²dγ

d² =−2(divv)nl^>n+ (divv)²−tr(l l) + 2nl^>l^>n

+ div ˙v−ndv˙^>n+nl l^>n−(nl^>n)²dγ. (5.109) In surface-adapted curvilinear coordinates, equation (5.109) simplifies to

d²dγ

d² =^h(divγv)²−(nl^>n)²−tr(l l) + 2nl^>l^>n+nl l^>n−(nl^>n)²+ divγv˙ⁱdγ.

(5.110) Using index notation, we can simply various terms in expression (5.110) and group them into contributions intrinsic and extrinsic to the surfaceγ,

2nl^>l^>n−tr(l l)−(nl^>n)² = 2(∇3vⁱ)(∇iv³)−(∇jvⁱ)(∇iv^j)−(∇3v³)²

= 2(∇3v³)(∇3v³) + 2(∇3v^α)(∇αv³)−(∇3v³)(∇3v³)

−(∇αv³)(∇3v^α)−(∇βv^α)(∇αv^β)−(∇3v^α)(∇αv³)

−(∇3v³)²

=−(∇βv^α)(∇αv^β) =−( ˆ∇βv^α−ii^α_βv³)( ˆ∇αv^β−ii^β_αv³)

=−( ˆ∇βv^α)( ˆ∇αv^β) + 2ii^α_βv³( ˆ∇αv^β)−ii^β_αii^α_β(v³)², (5.111) nl l^>n−(nl^>n)² = (∇jv³)(∇^jv₃)−(∇3v³)²

= (∇αv³)g^αβ(∇βv₃)

= ( ˆ∇αv³+ii_αα0v^α0)g^αβ( ˆ∇βv₃+ii^β_β⁰v_β0)

= ( ˆ∇αv³)( ˆ∇βv₃) +ii^β_α0v^α0( ˆ∇βv₃)

+( ˆ∇αv³)ii^αβ0v_β0+ii^β_α0ii^β_β⁰v^α0v_β0, (5.112) (divγv)² = (divγv_γ)²−4h(divγv_γ)vn+ 4h²v_n², (5.113) where∇ˆα is the low-dimensional covariant derivative restricted to surfaceγ, for which

∇βv^α = ˆ∇βv^α−ii^α_β on γ (5.114)

holds (recall expansions (4.29) of the first kind Christoffel symbol). Substituting ex- pressions (5.111), (5.112) and (5.113) into second total derivative (5.110) of dγ and exploiting properties of the second fundamental formii we arrive at

d²|dγ|

d² =^hdivγv˙ + (divγv)²−(∇βv^α)(∇αv^β) + (∇αv³)(∇^αv³)ⁱdγ

=^hdivγv˙ + (divγv_γ)²−4h(divγv_γ)vn+ 4h²v_n²

−( ˆ∇βv^α)( ˆ∇αv^β) + 2ii^α_βv³( ˆ∇αv^β)−ii^β_αii^α_β(v³)² +( ˆ∇αv³)g^αβ( ˆ∇βv₃) + 2ii^β_α0v^α0( ˆ∇βv₃) +ii^β_α0ii^β_β⁰v^α0v_β0

idγ

=^hdivγv˙ + (divγv_γ)²−4h(divγv_γ)vn+ 4h²v_n²

−( ˆ∇βv^α)( ˆ∇αv^β) + 2ii^α_β(v³∇ˆαv^β +v^β∇ˆαv³)−(4h²−2k)(v³)² +( ˆ∇αv³)( ˆ∇^αv₃) +ii^β_α0ii^β_β⁰v^α0v_β0

idγ

=^hdivγv˙ + (divγv_γ)²−( ˆ∇βv^α)( ˆ∇αv^β) + ( ˆ∇αv_n)( ˆ∇^αv_n) + 2kv²_n

−4h(divγv_γ)vn+ 2ii^α_β∇ˆα(vnv^β) +ii^β_α0ii^β_β⁰v^α0v_β0

idγ (5.115)

where∇ˆαis the low-dimensional covariant differential operation of surfaceγ. A cleanup of the algebras in expression (5.115) leads to the final result

d²dγ

d² =ⁿdivγv˙

+ 2hvγiivγ−kvγ·vγ

+dγvn·dγvn+ 2kv²_n

+ (divγv_γ)²−( ˆ∇βv^α)( ˆ∇αv^β)

−4h(divγv_γ)vn+ 2ii^α_β∇ˆα(vnv^β)^odγ.

(5.116)

The first line in (5.116) is due to spatial acceleration v˙ which is expected to have the identical form with the first derivative d(dγ)/d in (5.104) with v replaced by v.˙ The bilinear form in the second line in (5.116) is known as the third fundamental form tensor iii_αβ = 2hii_αβ −kg_αβ. Integrands from the third and the fourth lines are the intrinsic contributions from tangential vector field v_γ and from scalar (normal) field v_n, respectively. We interpret the geometric meaning of these intrinsic contributions by recognizing

(divγvγ)²−( ˆ∇βv^α)( ˆ∇αv^β) = (divγvγ)²−tr (dγvγ)(dγvγ)= 2 det( ˆ∇βv^α). (5.117) Expression (5.117) exactly agrees with the second derivative of Jacobian∂²J/∂² from (5.30) in the case of a flat two-dimensional manifold with tangential deformation v_γ only. The last line in (5.116) represents the three-way coupling between v_γ, v_n and the extrinsic (mean) curvature of surfaceγ. If the second derivative ofdγ in (5.117) is

restricted to normal variations only (i.e. v = v_nn), we then arrive at the well known formula in shape analysis,

d²dγ

d² =divγv˙ +|dγvn|²+ 2kv_n²dγ when vγ =0. (5.118) In the theory of minimal surfaces, integration of expression (5.118) over the entire surface of a critical shape γ leads to a quadratic shape functional whose convexity directly determines whether its total surface area is locally minimal or maximal. The expression of the first area variation (5.104) and the second variation (5.118) restricted to normal displacement can be found in standard textbooks on differential geometry (Stoker, 1988; Kreyszig, 1991). However, the full expression (5.116) of second area variation d²dγ/d² especially including the tangential variations of the surface is less commonly known or documented, at least to the best of the author’s knowledge. We find it coincides with the expression derived in Capovilla and Guven (2004).

First and second derivatives of total volume

Variations of volume differential element are straightforward since they mostly deal with the relative Jacobian J. The first derivative of spatial volume differentialdω is directly given by ∂J/∂in (5.29),

ddω d = ∂J

∂ dΩ= divvdω. (5.119)

The first derivative of the total volume can be expressed as a surface integral through divergence theorem,

d d

Z

ωdω=^Z

Ω

∂J

∂ dΩ=^Z

γ

v·ndγ =^Z

γ

v_ndγ. (5.120)

The second derivative of spatial volume differential dω is directly given by ∂²J/∂² in (5.30),

d²|dω| d² = ∂²J

∂² dΩ=div ˙v+ (divv)²−tr(l l)dω. (5.121) In order to convert the second derivative of the total volume

d² d²

Z

ωdω=^Z

Ω

∂²J

∂² dΩ=^Z

ωdiv ˙v+ (divv)²−tr(l l) dω (5.122) to a surface integral only, we first recall an identity similar to the one we encounter in (5.76),

div(divv)v−lv=∇i(∇jv^j)vⁱ−(∇jvⁱ)v^j

= (∇jv^j)(∇ivⁱ)−(∇jvⁱ)(∇iv^j) + (∇i∇jv^j)vⁱ−(∇i∇jvⁱ)v^j

= (divv)²−tr(l l). Flat (5.123)

With identity (5.123) substituted in (5.122), the second derivative of the total volume now becomes a surface integral,

d² d²

Z

ωdω=^Z

γ

n·v˙ + (divv)(n·v)−nlvdγ. (5.124) We can gain more insights into the second derivative (5.124) by expanding it using index notation of surface-adapted curvilinear coordinates,

(divv)(n·v)−nlv= (∇ivⁱ)v³−(∇iv³)vⁱ = (∇αv^α)v³−(∇αv³)v^α

= ( ˆ∇αv^α−ii^α_αv³)v³−( ˆ∇αv³+ii_αβv^β)v^α

=v³∇ˆαv^α−v^α∇ˆαv³−2h(v³)²−ii_αβv^βv^α. (5.125) Finally with identity (5.125) we arrive at

d² d²

Z

ωdω=^Z

γv˙n+vn(divγvγ)−vγ·dγvn−2hv²_n−vγiivγdγ. (5.126) As usual, the first term in the second derivative of the total volume (5.126) is due to spatial accelerationv. The next two terms are the intrinsic coupling between tangential˙ vector fieldv_γ and scalar normal fieldv_n, The last two terms represent how vector field vγ and scalar field vn individually interact with extrinsic curvatures of surfaceγ. Both first (5.120) and second (5.126) derivatives of the total volume are surface integral and only involve information of vector fieldvdefined strictly intrinsic to surfaceγ (e.g., n·dv is considered extrinsic to surface γ). This is expected because the total volume of region ω can be expressed as a surface integral via divergence theorem,

Z

ωdω=^Z

ω

1

3divxdω= 1 3

Z

γn·xdγ. (5.127)

Finally we would like to highlight the necessity of spatial acceleration v˙ in volume conservation. Consider a configuration mapχ(·, )which is constrained to conserve the total volume for all valid. The total derivative of ^R_ωdω with respect to at any order must be zero over the entire range ofas well. As implied by (5.120), a spatial velocity field v = v_γ always tangent to surface γ would guarantee that the first derivative of R

ωdωvanishes but not necessarily the second derivative! To see this claim, let’s consider a unit sphere to be the material frameΩwithχ(X, )being a rigid rotation that rotates Ω to a new spatial frameω about some axis (in this case is the rotation angle). We expect the total volume of spatial domain ω be conserved for all . Since v derived from χ(X, ) is everywhere tangent to the surface γ of the sphere, the first derivative of ^R_ωdω is zero according to formula (5.120). Second fundamental formii in this case equals to the negative of the identity tensor everywhere on the surface of a unit sphere which renders the second derivative of the total volume

d² d²

Z

ωdω=^Z

γv˙_n+v_γ·v_γdγ (5.128)

through the formula (5.126). It’s now evident that, integral (5.128) would always be non-negative without the help of spatial acceleration v˙ and hence violate volume con- servation.