Poincar\'e Path Integrals for Elasticity

Poincar´e Path Integrals for Elasticity

arXiv:1801.07058v1 [math-ph] 22 Jan 2018

Snorre H. Christiansen∗∗

Kaibo Hu††

Espen Sande‡‡

Abstract We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators P for elasticity satisfying DP + PD = id and P 2 = 0, where the differential operators D correspond to the linearized deformation, the linearized curvature and the divergence, respectively. As a special case, we rederive the classical Ces` aro-Volterra path integral for strain tensors satisfying the Saint Venant compatibility condition, using the path integral for scalar potentials and techniques from homological algebra. The known path-independence of the Ces` aro-Volterra path integral can thus be seen as a result of the corresponding property of differential 1-forms. In general we derive path integral formulas in the presence of defects.

1

Introduction

In the classical theory of linear elasticity, a second order symmetric tensor satisfying the Saint Venant compatibility condition inc e = 0 on a contractible domain, is a symmetric gradient (linearized strain tensor) of some displacement vector field u, i.e. e = def u. Here, inc e := ∇ × e × ∇ is called the
incompatibility and def u := 1/2 ∇u + ∇T u is the deformation. In fact, the displacement field can be recovered from an explicit path integral. This is a classical result that dates back to the work of Ces`aro in 1906 and Volterra in 1907 [13, 31]: Z (eij (y) + (∂k eij (y) − ∂i ekj (y)) (xk − yk )) dyj , (.) ui (x) = γ(x)

or equivalently in the vector form u(x) =

Z

e(y) + (x − y) ∧ (∇ × e(y)) dy.

γ(x)

Compared with path integrals for the de Rham complex, the right hand side of the Ces`aro-Volterra path integral (.) contains an additional derivative term ∇ × e, called the Frank tensor (c.f. [30]). On simply connected domains, the integral (.) does not depend on the chosen path between fixed end points. ∗∗ Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO 0316 Oslo, Norway. email:[email protected] †† Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO 0316 Oslo, Norway. email: [email protected]. ‡‡ Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO 0316 Oslo, Norway. email:[email protected]

1

There has been a lot of recent progress on the Ces`aro-Volterra path integral. Non-simply-connected bodies were considered in [32]. Generalizations to weaker regularity and on surfaces were given in [18] and [19], respectively. Geometric reductions for the plate models were derived in [23] based on asymptotic expansions of the Ces` aro-Volterra integral. The Frank tensor appearing in the integral can be used as the boundary term in the intrinsic elasticity formulation [30, 17]. This formulation uses the strain tensor as the major variable, and the displacement can be recovered by the Ces`aro-Volterra path integral. We refer to [16, 17, 30] for progress in this direction. With the same second order differential operator inc, for a divergence-free symmetric stress tensor σ, one can find a symmetric matrix field w such that σ = inc w. This w is called the Beltrami stress function [24]. More generally, defects can be regarded as a microscopic mechanism of plastic deformations [27]. When defects are present in crystal materials, the displacement is not continuous and as a continuum theory, the strain tensor does not satisfy the Saint Venant compatibility condition anymore. In this case, the incompatible strain has the Beltrami decomposition e = def u + inc r,

(.)

where r is a divergence-free symmetric matrix field. The Beltrami decomposition (.) has been used in [2] to derive the elasto-plasticity model, instead of the additive decomposition e = ee + ep , where the plastic strain ep is related to the dislocation density κ by the Kr¨oner relation (c.f. [28, 1]) inc ep = −κ × ∇.

More generally, properties of the strain and stress tensors can be expressed in terms of the 3D elasticity complex 0

✲ RM

⊆ ✲ C ∞ (Ω; V) def ✲ C ∞ (Ω; S) inc ✲ C ∞ (Ω; S) div ✲ C ∞ (Ω; V)

✲ 0.

(.)

where C ∞ (Ω; V) and C ∞ (Ω; S) denote the spaces of smooth vector- and symmetric-matrix-valued functions, respectively. Kr¨oner is one of the pioneers relating the incompatibility inc e with the defect densities [27, 28, 29], and therefore (.) is also referred to as the Kr¨oner complex in the literature. The elasticity complex (.) can be related to the standard de Rham complex by the Bernstein-GelfandGelfand (BGG) construction [20, 21, 12]. Arnold, Falk and Winther [6, 9] introduced the BGG to numerical analysis and used it to construct finite elements for the elasticity complex, via the de Rham complex. The goal of this paper is to use the BGG to construct path integrals for elasticity. The proof of the Poincar´e lemma, i.e. that the de Rham complex is locally exact, is based on the operators p satisfying the null-homotopy formula dp + pd = id,

(.)

where d is the exterior derivative and id is the identity operator. Particularly, given a closed form u (satisfying du = 0), we immediately obtain u = d(pu) from (.). This implies that u is also exact, and that pu is a potential for u. If we choose a path γ : t 7→ tx, t ∈ [0, 1], connecting a base point (chosen

2

to be the origin here) and an arbitrary point x, then the Poincar´e operator acting on a k-form reads Z 1 (pu)x := tk−1 utx (x, ·)dt. (.) 0

Here ux is the evaluation of u at the point x and u(x, ·) is the contraction of u with the tangent vector x. Note that for k = 1, the integral (.) does not depend on the path γ between fixed end points when du = 0. In addition to the homotopy relation (.), the operators p also satisfies the following properties: (i) the complex property: p2 = 0; (ii) the polynomial preserving property: if u is a homogeneous polynomial of degree r, then pu is a homogeneous polynomial of degree r + 1. This reflects that the differential operators in the de Rham complexes are first order. We remark that these operators also play an important role in the construction of finite element methods [25, 8, 15]. In this paper, we propose a methodology to derive Poincar´e type operators for differential complexes based on the BGG construction and the classical Poincar´e operators for the de Rham complex. In particular, we derive explicit path integral formulas for the elasticity complex (.). This rather different approach yields the classical Ces`aro-Volterra integral as a special case. Furthermore, these explicit Poincar´e operators for elasticity satisfy the properties analogous to those mentioned for the de Rham complex. As a result, we generalize the Ces`aro-Volterra integral in the presence of defects. The presentation of the BGG construction here will mainly follow the conventions in the numerical analysis literature [6, 22]. The remaining part of this paper will be organized as follows. In Section 2 we recall definitions and preliminary results on the de Rham complex and the elasticity complex. In Section 3 we present the Poincar´e and Koszul operators for 2D and 3D elasticity complexes, which are new in the literature to our knowledge. In Section 4 we review the derivation of the elasticity complex from the de Rham complex via the BGG construction. In Section 5 we propose a new methodology to derive Poincar´e operators based on the BGG construction and derive the operators for the elasticity complex. We give some concluding remarks in Section 6.

2

Notation and Preliminaries

We use V to denote the space of vectors in Rn and use M to denote the space of n × n matrices and S, K for the subspaces of symmetric and skew-symmetric matrices respectively. We further define the product space W := K × V. We use Λk (Ω; W) to denote the space of smooth W- valued kforms. Similarly we can define symmetric and skew-symmetric matrix valued differential forms. In the vector proxy notation, this means that the coefficients in front of the form basis are matrices (symmetric or anti-symmetric matrices). The kernel of the linearized deformation operator, RM := {u = a + b ∧ x : a, b ∈ V} is called the space of rigid body motions. We denote the vector x = (x1 , x2 ) and x⊥ = (x2 , −x1 ) in 2D and x = (x1 , x2 , x3 ) in 3D. We use Pr (M) to denote the space of matrix valued polynomials of degree at most r, and use Hr (M) to denote the space of homogeneous polynomials of degree r, i.e. q ∈ Hr (M) implies q(tx) = tr q(x) 3

where t ∈ R is a scaling factor. We also define similar spaces for symmetric matrices S and skewsymmetric matrices K. Recall the 2D differential operators ! ! ∂u1 −∂2 v ∂u2 u1 − , and curl v = . = rot ∂x1 ∂x2 ∂1 v u2 We rarely use rot in this paper. The notation ∇× denotes the curl operators both in 2D and 3D. In 2D, the curl maps a scalar to a vector while in 3D the curl operator maps a vector to a vector. For u ∈ C ∞ (M), it is important to distinguish curl operators on the left and on the right: ∇ × u is to take curl on each column while u × ∇ is to take curl on each row. Using the index notation, this means (∇ × u)ij = ǫi ab ∂a ubj and (u × ∇)ij = ǫjab ∂a uib where the Einstein summation convention has been used. As a standard notation, we use u ⊗ v to denote the tensor product of the vectors u and v, i.e. (u ⊗ v)ij = ui vj . We use ! 0 −1 χ= 1 0 to denote the canonical skew-symmetric matrix in 2D. De Rham complexes, Poincar´ e and Koszul operators ential k-forms on a manifold Ω. The complex 0

✲ R

d0✲

✲ Λ0 (Ω)

Λ1 (Ω)

Let Λk be the space of smooth differ-

d1 ✲ ···

dn−1 ✲

Λn (Ω)

✲ 0

is called the de Rham complex. Here di is the ith exterior derivative and n is the dimension of the manifold Ω. The exterior derivative satisfies the complex property di+1 di = 0, for i = 1, ..., n−1. When it is clear from the context we will drop the index of d. The complex property can then be written as d2 = 0. We consider a smooth map F : [0, 1] × Ω 7→ Ω, and interpret it as a family of maps Ft = F (t, ·) : Ω 7→ Ω, for t ∈ [0, 1], defining a homotopy between F0 and F1 . For t ∈ [0, 1], we assume there is a vector field Gt on Ω such that, for x ∈ Ω: Gt (Ft (x)) = ∂t Ft (x).

(.)

This uniquely defines Gt on Ω whenever Ft : Ω 7→ Ω is a diffeomorphism, and expresses that any curve F• (x) flows with G• . The Poincar´e operator associated with F (and G), acting on differential k-forms, is denoted p[F ] or, when the choice of F is clear, as p. It can be written succinctly as: Z 1 (.) p[F ]u = Ft⋆ (iGt u) dt. 0

More explicitly, if u is a k-form: (p[F ]u)x (ξ2 , . . . , ξk ) =

Z

1

uFt (x) (∂t Ft (x), DFt (x)ξ2 , . . . , DFt (x)ξk ) dt.

(.)

0

From Cartan’s formula for Lie derivatives, it follows that the Poincar´e operator has the property: F1⋆ − F0⋆ = p[F ]d + dp[F ]. 4

(.)

Suppose that F1 is the identity on S and that F0 is constant. Then the formula gives, for p = p[F ] acting on k-forms with k ≥ 1: id = pd + dp, (.) whereas if u is a function, considered as a 0-form, and the value of F0 is W , we get: u − u(W ) = pdu.

(.)

If now Ω is a domain in an affine space, which is starshaped with respect to, say W , we may choose F to be defined by: Ft (x) = tx + (1 − t)W. (.) Then we may substitute in the above formulas: ∂t Ft (x) = x − W, Gt (y) =

1 (y − W ), and DFt (x)(ξ) = tξ. t

(.)

We denote the associated Poincar´e operator as pW . It is defined explicitly on k-forms u by: Z

(pW u)x (ξ2 . . . , ξk ) =

1

tk−1 uW +t(x−W ) (x − W, ξ2 , . . . , ξk ) dt.

(.)

0

In an affine space, given a choice of a point W , we may also define directly a vector field XW by: XW : x 7→ x − W.

(.)

In vector form, the 3D Poincar´e operators in an affine space with W = 0 read: Z 1 p1 u = utx · xdt, ∀u ∈ C ∞ (V), 0

p2 v =

1

Z

∀v ∈ C ∞ (V),

tvtx ∧ xdt,

0

p3 w =

Z

1

t2 wtx xdt,

∀w ∈ C ∞ (R),

0

which satisfy

p1 grad φ = φ + C,

∀φ ∈ C ∞ (R),

p2 curl u + grad p1 u = u,

∀u ∈ C ∞ (V),

p3 div w + curl p2 w = w,

∀w ∈ C ∞ (V).

(.)

Here C = −φ(0) in (.) indicates that the identity p1 grad φ = φ holds up to a constant. The contraction of a differential form by XW is called the Koszul operator associated with W and denoted: (.) κW : u 7→ κW u = iXW u. If the choice of W is clear from the context, we may sometimes omit it from the notation. The Koszul operators can be used to simplify the construction of some classical finite elements [8].

5

If u is a k-form which, with respect to some choice of origin W and basis of Rn , has components that are homogeneous polynomials of degree r, then from (.) we get: pW u =

1 κW u, k+r

(.)

which is a polynomial whose components are homogeneous of degree r + 1. From the above discussion of Poincar´e operators, we can derive the identity, on k-forms which are homogeneous polynomials of degree r: (dκW + κW d)u = (r + k)u.

(.)

Lastly, we remark that both the Poincar´e operators and the Koszul operators satisfy the complex property: p2W = κ2W = 0, for the choice (.). Elasticity complexes Let Ω be a contractible domain in 2D. The elasticity complex in 2D reads 0

✲ P1

⊆✲ ∞ airy div C (Ω) ✲ C ∞ (Ω; S) ✲ C ∞ (Ω; V)

✲ 0.

(.)

The operator airy : C ∞ (R) 7→ C ∞ (S) defined by airy(u) := ∇×u×∇ in 2D is called the Airy operator, which is a rotated version of the Hessian: ! ∂22 u −∂1 ∂2 u airy(u) = . −∂1 ∂2 u ∂12 u In elasticity theory, the Cauchy stress is a second order symmetric tensor, appearing in C ∞ (Ω; S) in the sequence (.). The divergence operator, defined row-wise, maps onto C ∞ vectors and the kernel can be parametrized by C ∞ scalar functions through the Airy operator. The complex (.) describes the kinetics of the elasticity, i.e. the Cauchy stress and the force, while the following complex characterizes the kinematics, i.e. the geometry of deformation: 0

✲ RM

⊆✲ ∞ def rot rot ✲ C ∞ (Ω; V) C (Ω) ✲ C ∞ (Ω; S)

✲ 0.

(.)

The complex (.) is essentially dual to (.) with proper boundary conditions and the L2 inner product. The above sequence involves the linearized deformation operator def and the strain tensor in planar elasticity. Now, let Ω be a contractible domain in R3 . Then the elasticity complex has the form of (.). As in the 2D case, the last two spaces characterize the Cauchy stress and the force vector (kinetics). The first several spaces of the 3D sequence are analogous to (.). The space C ∞ (Ω; V) describes the displacement in linear elasticity, and def is the linearized deformation. The second space C ∞ (Ω; S) includes strain tensors (symmetric second order tensors) and the operator inc is the linearized curvature tensor. By straightforward calculations we see inc(def(u)) = 0, which shows that in the classical theory of linear elasticity any strain tensor G obtained from a displacement field satisfies inc(G) = 0. Due to the exactness of the complex in R3 , any symmetric tensor G satisfying inc G = 0 is the linearized strain tensor of some displacement field. The condition inc G = 0 is called the Saint-Venant compatibility condition in linear elasticity.

6

The complex (.) is self-dual in the sense that for C0∞ functions u Z Z def(u) : σ dx = − u · div(σ) dx, ∀u ∈ C0∞ (Ω; V), σ ∈ C ∞ (Ω; S), Ω

and

(.)

Ω

Z

inc g : τ dx =

Z

∀g ∈ C0∞ (Ω; S), τ ∈ C ∞ (Ω; S).

g : inc τ dx,

Ω

Ω

The identity (.) also holds for u ∈ C ∞ (Ω; V) and σ ∈ C0∞ (Ω; S). The 3D de Rham complex has a similar property. For the 2D case, the complexes (.) and (.) are formally dual to each other. Calabi [11] constructed a complex on manifolds with constant sectional curvature. The operators are the Lie derivatives of the metric, the linearized curvature and the divergence respectively. In the Euclidean space R3 , the Calabi complex considered in [11] degenerates to (.) (c.f. [4, 3, 26]). An application of the complex (.) to the Regge calculus, a discrete theory of general relativity, can be found in Christiansen [14].

3

Main results: Poincar´ e and Koszul operators for the elasticity complex

In this section we summarize the main results for the elasticity complex. We remark again that the derivation of Poincar´e operators from the BGG construction is, as far as we know, a new methodology and the following results for the elasticity complex are a particular example of this approach. Poincar´ e operators Theorem 1 (2D case). Assume Ω = R2 . We define P1 : C ∞ (S) 7→ C ∞ (R) by Z 1 (1 − t)x⊥ · utx · x⊥ dt. P1 : u 7→ 0

and define P2 : C ∞ (V) 7→ C ∞ (S) by Z 1 Z P2 : u 7→ sym tutx ⊗ x dt + 0

1

0

 t(t − 1)(x · utx )x dt × ∇ , ⊥



where the 2D scalar curl operator “×∇” maps each component of the vector sym  R1 ⊥ 0 t(t − 1)(x · utx )x dt to a row vector. Then we have P1 (airy u) = u + P1 ,

P2 div µ + airy P1 µ = µ,

∀u ∈ C ∞ (Ω),

R1 0

tutx ⊗ x dt +

(.)

∀µ ∈ C ∞ (Ω; S),

and div P2 v = v,

∀v ∈ C ∞ (Ω; V),

where P1 in (.) indicates that the identity P1 airy u = u holds up to P1 , the kernel of airy. Given u, the linear function in (.) can be expressed by u and its derivatives (1-jets) at the origin by P2 u0 + i=1 (∂i u0 )xi . Particularly, for a matrix field µ satisfying div µ = 0, we can find a scalar function φ := P1 µ satisfying airy φ = µ. For any vector function v, we can explicitly find a matrix potential ψ := P2 v satisfying div ψ = v. 7

Theorem 2 (3D case). Let Ω = R3 . We define P1 : C ∞ (Ω; S) 7→ C ∞ (Ω; V) by P1 (ω) :=

Z

1

ωtx · xdt +

0

1

Z

(1 − t)x ∧ (∇ × ωtx ) · xdt,

0

define P2 : C ∞ (Ω; S) 7→ C ∞ (Ω; S) by P2 : µ 7→ x ∧

Z

1



t(1 − t)µtx dt ∧ x, 0

and define P3 : C ∞ (Ω; V) 7→ C ∞ (Ω; S) by P3 : µ 7→ sym

Z

1

2

t x ⊗ µdt −

0

Z

1

0

 t (1 − t)x ⊗ µ ∧ xdt × ∇ .

(.)

∀u ∈ C ∞ (Ω; V),

(.)

2



Then we have P1 (def u) = u + RM, P2 inc µ + def P1 µ = µ,

∀µ ∈ C ∞ (Ω; S),

P3 div ω + inc P2 ω = ω,

∀ω ∈ C ∞ (Ω; S),

and div P3 v = v,

∀v ∈ C ∞ (Ω; V),

where RM in (.) indicates that the identity P1 (def u) = u holds up to the rigid body motion, i.e. the kernel of def. Given u, the rigid body motion in (.) can be expressed by u and its rotation at the origin by u0 + 1/2(curl u)0 ∧ x. Particularly, for a symmetric matrix valued function µ satisfying inc µ = 0, we have (Ces` aroVolterra path integration) µ = def (P1 µ) . For a symmetric matrix valued function ω satisfying div ω = 0, we have ω = inc (P2 ω) . The above integrals are with respect to a special path γ : t 7→ tx, connecting a base point 0 with x. Since the Poincar´e operators for the de Rham complex can be defined along an arbitrary path, we can also derive corresponding operators for the elasticity complex on a general path by repeating the BGG steps. Observe that by choosing the special path γ in the Cesar`o-Volterra formula (.) we see that it coincides with the operator P1 . Theorem 3. The Poincar´e operators derived above satisfy the complex property P 2 = 0. Proof. For the 2D case, we find from a straightforward calculation that:

This implies that

  [(x⊥ · u)x] × ∇ = x ⊗ (x⊥ · u) × ∇ + (x⊥ · u)χ.
x⊥ · sym [(x⊥ · u)x] × ∇ · x⊥ = 0, 8

and P1 P2 = 0. For the 3D case, we find from another straightforward calculations that: (x ⊗ µ ∧ x) × ∇ = 3x ⊗ µ − µ ⊗ x + x ⊗ ∇x µ − (∇ · µ)x ⊗ x.

(.)

Due to this and the fact that x ∧ x = 0, we have the identity P2 P3 = 0. Lastly, we find that ∇ × (x ∧ u ∧ x) = −3u ∧ x − (x · ∇)u ∧ x + x ⊗ ((div u) ∧ x) + x ⊗ vec skw u, which implies that ∇ × (x ∧ u ∧ x) · x = 0 if u is symmetric. Here vec = Skw−1 : C ∞ (K) 7→ C ∞ (V) is the canonical identification between a vector and a skew-symmetric matrix (see also (.) below) and skw : C ∞ (M) 7→ C ∞ (K) defined by skw v = 1/2(v − v T ) is the skew-symmetrization operator. Therefore P1 P2 = 0.

Theorem 4. The Poincar´e operators defined above are polynomial preserving: • in 2D, u ∈ Pr (S) ⇒ P1 u ∈ Pr+2 (R),

v ∈ Pr (V) ⇒ P2 v ∈ Pr+1 (S);

• in 3D, u ∈ Pr (S) ⇒ P1 u ∈ Pr+1 (R),

v ∈ Pr (S) ⇒ P2 v ∈ Pr+2 (S),

w ∈ Pr (V) ⇒ P3 v ∈ Pr+1 (S).

Analogous to the de Rham case, the sequence 0✛

RM ✛

P1 P2 C ∞ (Ω; V) ✛ C ∞ (Ω; S) ✛ C ∞ (Ω) ✛

0

(.)

in two space dimensions, and the sequence 0✛

RM ✛

P1 P2 P3 C ∞ (Ω; V) ✛ C ∞ (Ω; S) ✛ C ∞ (Ω; S) ✛ C ∞ (Ω; V) ✛

0

(.)

in three space dimensions are complexes since P 2 = 0 holds. Furthermore, by the homotopy relation, both (.) and (.) are exact if Ω is star shaped. Koszul operators Analogous to the de Rham case we derive Koszul operators for the elasticity complex by applying the above Poincar´e operators on homogeneous polynomials of degree r. Theorem 5 (2D Koszul operators). Define K1r : C ∞ (S) 7→ C ∞ (R) by K1r : u 7→

1 x⊥ · utx · x⊥ , r+2 9

and define P2 : C ∞ (V) 7→ C ∞ (S) by K2r : u 7→ sym (u ⊗ x) − Then we have

  1 sym (x⊥ · utx ⊗ x) × ∇ . r+3

K1r−2 airy u = (r − 1)u + P1 ,

∀u ∈ Hr (R), r ≥ 2,

K2r−1 div σ + airy K1r σ = (r + 1)σ, div K2r w = (r + 2)w,

∀v ∈ Hr (S),

∀v ∈ Hr (V).

Theorem 6 (3D Koszul operators). Define K1r : C ∞ (V) 7→ C ∞ (S) by K1r : ω 7→ ω · x +

1 x ∧ (∇ × ω) · x, r+2

∀ω ∈ C ∞ (S),

and K2r , K˜2r : C ∞ (S) 7→ C ∞ (S): K2r : u 7→

1 x ∧ u ∧ x, r+2

∀u ∈ C ∞ (S),

K˜2r : u 7→

1 x ∧ u ∧ x, r+3

∀u ∈ C ∞ (S),

and define K3r : C ∞ (V) 7→ C ∞ (S) by: K3r : v 7→ sym(x ⊗ v) −

1 sym ((x ⊗ v ∧ x) × ∇) , r+4

∀v ∈ C ∞ (V).

Then we have K1r−1 def u = r · u + RM,

∀u ∈ Hr (V),

def K1r v + K2r−2 inc v = (r + 1)v, inc K˜2r w + K3r−1 div w = (r + 2)w, div K3r µ = (r + 3)µ,

∀v ∈ Hr (S), ∀v ∈ Hr (S),

∀µ ∈ Hr (V).

Similar to the Poincar´e operators (Theorem 3), the Koszul operators satisfy the complex property. Theorem 7. The Koszul operators derived above satisfy the complex property K sense. In 2D, we have K1r+1 K2r = 0, ∀r = 0, 1, · · · ,

2

= 0 in the following

In the 3D case, we have K1r+2 K2r = 0,

∀r = 0, 1, · · · ,

K2r+1 K3r = 0,

∀r = 0, 1, · · · .

and

Theorem 8. The Koszul operators defined above are polynomial preserving: • in 2D, u ∈ Hr (S) ⇒ K1r u ∈ Hr+2 (R),

10

v ∈ Hr (V) ⇒ K2r v ∈ Hr+1 (S);

• in 3D, u ∈ Hr (S) ⇒ K1r u ∈ Hr+1 (R),

v ∈ Hr (S) ⇒ K2r v ∈ Hr+2 (S),

w ∈ Hr (V) ⇒ K3r v ∈ Hr+1 (S).

This implies that the sequences 0✛

RM ✛

K r+1 Kr Hr+3 (Ω; V) ✛ 1 Hr+1 (Ω; S) ✛ 2 Hr (Ω) ✛

0

(.)

in two space dimensions, and 0✛

RM ✛

K r+3 K r+1 Kr Hr+3 (Ω; S) ✛ 2 Hr+1 (Ω; S) ✛ 3 Hr (Ω; V) ✛ Hr+4 (Ω; V) ✛ 1

0 (.)

in three space dimensions are complexes. By the homotopy relation, both (.) and (.) are exact if Ω is contractible. Duality We recall the Koszul operators for the de Rham complexes (in the 3D case as an example): κ1 : u 7→ u · x,

∀u ∈ C ∞ (V),

κ2 : v 7→ v ∧ x,

∀v ∈ C ∞ (V),

κ3 : w 7→ wx,

∀w ∈ C ∞ (R).

They have the duality (κ1 u) · v = u · (κ3 v), v · (κ2 w) = −w · (κ2 v),

∀u ∈ C ∞ (V), v ∈ C ∞ , ∀u ∈ C ∞ (V), v ∈ C ∞ (V).

(.) (.)

For the Koszul operators derived above, we observe an analogous relation of duality. Specifically, we have x ∧ u ∧ x : v = u : x ∧ v ∧ x, which implies that K2r u : v = u : K2r v. Moreover, K1r u · v = v · u · x +

1 v · (x ∧ ∇ × u) · x, r+2

and u : K3r v = u : sym(x ⊗ v) −

1 u : (x ⊗ v ∧ x) × ∇. r+4

We note the identities u : sym(x ⊗ v) = v · u · x, and the integration by parts Z Z Z u : [(x ⊗ v ∧ x) × ∇] = −(u × ∇) : (x ⊗ v ∧ x) = − x · (u × ∇) · (v ∧ x) Z Z = x · (u × ∇) ∧ x · v = − v · (x ∧ (∇ × u)) · x, 11

which holds for u with certain vanishing conditions on the boundary of the domain, e.g. u ∈ C0∞ (S). This implies that Z Z K1r+2 u : v = u : K3r v. Remark 1. Compared with the Koszul operators for the de Rham complexes, the definitions given in Theorem 6 contain correction terms involving derivatives (curl operators in 3D) and these terms explicitly depend on the degree of the homogeneous polynomials. As we have seen, these terms together with the polynomial degree naturally match with each other in the null-homotopy formulas (only noting the modification in K˜2r due to the second order differential operators inc) and in the duality. As discussed in the introduction for the classical Cesar` o-Volterra formula, these derivative terms have physical significance in materials with incompatibility.

4

Bernstein-Gelfand-Gelfand Construction

The Bernstein-Gelfand-Gelfand (BGG) construction was originally developed in the theory of Lie algebras [10]. Relations with the de Rham complexes and a derivation of the elasticity complex can be found in Eastwood [21, 20]. In this section, we recall the derivation of the elasticity complex from the de Rham complex by the BGG construction. The BGG construction for the 2D and 3D elasticity complexes can be summarized in (.) and (.) respectively.

12

W

Λ0 (W)

A

(ω, µ) ↓ (ω, S0−1 dω) W

Γ0

A

(ω, S0−1 dω) ↓ ω

W

Λ0 (K)

dS0−1 d

J0

W

C ∞ (R)

C ∞ (R)

(ω, µ) ↓ (0, µ + dS0−1 ω) A

Γ1 (0, µ) ↓ µ

Λ1 (V)

airy

(−S1 , d)T

id

Λ2 (W)

Λ2 (W)

(skw, div)T ∞ C ∞ (M) C (K × V)

C ∞ (S)

13

0

0

0

J2

0

(ω, µ) ↓ µ − div ω

sym

airy

Λ2 (W)

id

J1

id

W

A

Λ1 (W)

div

C ∞ (V)

0

(.)

A

Λ0 (W)

W

A

Λ0 (W)

Λ0 (W)

(d, −S0 )

C ∞ (V × K)

(grad, id)

(−S2 , d)T

Λ3 (W)

inc

0

J3

div

C ∞ (S)

0

(ω, µ) ↓ µ − div ω

sym

C ∞ (S)

0

id

J2

sym

def

C ∞ (V)

Λ3 (W)

(0, µ) ↓ µ

Λ2 (V)

0

id

curl S1−1 curl (skw, div) ∞ C ∞ (M) C ∞ (M) C (K × V)

(ω, µ) ↓ ω

V×K

A

Γ2

J1

J0

V×K

dS1−1 d

Λ1 (K)

Λ3 (W)

(ω, µ) ↓ (0, µ + dS1−1 ω)

(ω, S1−1 dω) ↓ ω

id

W

A

Γ1

A

Λ2 (W)

(ω, µ) ↓ (ω, S1−1 dω)

id

W

A

Λ1 (W)

C ∞ (V)

0 (.)

The starting point of the BGG construction is the W-valued de Rham complex dn−1

d

W −−−−→ Λ0 (Ω; W) −−−0−→ Λ1 (Ω; W) −−−−→ · · · −−−−→ Λn (Ω; W) −−−−→ 0,

(.)

where W := K × V. Each element in Λk (Ω; W) has two components, one is a skew-symmetric valued k-form and another is a vector valued k-form. Define Ak : Λk (Ω; W) 7→ Λk+1 (Ω; W) by ! dk −Sk Ak := , 0 dk i.e. for (ω, µ) ∈ Λk (Ω; W), A (ω, µ) := (dω − Sµ, dµ) with Sk : Λk (V) 7→ Λk+1 (K) (see (.)).

✲

✲ ✲ Λk−1 (V)

✲ Λk (V)

k+

(.)

S

S

k

1

1 k−

S ···

✲ ···

✲

d d ✲ Λk+1 (K) ✲ Λk+2 (K)

✲ Λk (K)

···

d ✲ Λk+1 (V) 14

✲ ···

To define the S operator [7, 22, 21], we first introduce K : Λk (V) 7→ Λk (K) defined by K = x ⊗ ω − ω ⊗ x. The operator K only acts on the coefficients of the alternating forms. For any k-form, K maps a vector coefficient to a skew-symmetric matrix coefficient. In the 2D case, we assume ω = (ω1 , ω2 )T . Then ! 0 x1 ω2 − x2 ω1 Kω = x ⊗ ω − ω ⊗ x = . − (x1 ω2 − x2 ω1 ) 0 This anti-symmetric matrix is usually identified with the scalar − (x1 ω2 − x2 ω1 ). Then the operator S is defined by Sk = dk K − Kdk . By definition, the identity dk+1 Sk + Sk+1 dk = 0,

(.)

holds. From (.) it is easy to see that Ak+1 Ak = 0, or A 2 = 0 in short. It turns out that the operators S are algebraic in the sense that no derivatives are involved. Furthermore, in 2D, the operator S0 is bijective and S1 is surjective while in 3D, S0 is injective, S1 is bijective and S2 is surjective. In subsequent discussions we take the 3D case as an example. The first rows of (.), i.e. 0

✲ W

✲ Λ0 (W)

A✲ 1 Λ (W)

A✲ 2 Λ (W)

A✲ 3 Λ (W)

✲ 0,

(.)

is a complex. The second step is to filter out some parts of the complex. We take the 3D elasticity complex (.) as an example. The space Λ2 (W) has two components, i.e. ω and µ. In elasticity µ corresponds to the stress tensor, therefore we want to filter out the ω component. Specifically, we consider the subspace of Λ2 (W): {(ω, µ) ∈ Λ2 (W) : ω = 0}. To find out the pre-image of the operator A restricted on this subspace, we notice that A (u, v) = (du−S1 v, dv) = (0, µ) if and only if v = S1−1 du. This gives the projection from the first row to the second: we keep Λ0 (W) and Λ3 (W) and project Λ1 (W) and Λ2 (W) to the corresponding subspaces. One can verify that the diagram commutes. The next step is an identification i.e. we identify (ω, S1−1 dω) with ω and (0, µ) with µ. This step also leads to a commuting diagram. Then we identify differential forms and exterior derivatives with vectors/matrices and differential operators. Such identifications are called the vector proxies [8]. The operators Jk provide vector/matrix representations of the differential forms. These representations are isomorphisms. We will give explicit forms of the vector proxies below. This leads to the elasticity complex with weakly imposed symmetry (the fourth row of (.)). The last step is to project the complex into the subcomplex involving symmetric matrices. Vector proxies in BGG By straightforward calculations, the vector proxy of the S operators in 2D can be summarized as follows: S0 (w1 , w2 )T = −w2 χdx1 + w1 χdx2 , 15

which is bijective with the inverse S0−1 (µ1 χdx1 + µ2 χdx2 ) = (µ2 , −µ1 )T . For w11 w21

w=

!

dx1 +

!

w12 w22

dx2 ,

we have S1 w = −(w11 + w22 )χdx1 ∧ dx2 , which is surjective. There are two possibilities for the vector proxies in 2D, i.e., let a 1-form u1 dx1 + u2 dx2 correspond to either (u1 , u2 ) or (−u2 , u1 ). The former gives the grad − rot complex while the latter gives the curl − div complex. Since our concern in this paper is the elasticity complex, we focus on the latter. We let the vector valued differential forms Λk (V), k = 0, 1, 2 given by ! ! ! ! w1 w12 w11 w1 dx1 ∧ dx2 , dx2 , dx1 + , w2 w22 w21 w2 correspond to w1 w2

!

,

w12 w22

−w11 −w21

!

The second order operator acting on differential forms ! ∂1 ∂2 u −1 dx1 + dS0 d(uχ) = −∂12 u

w1 w2

!

∂22 u −∂1 ∂2 u

!

,

.

dx2 ,

then corresponds to −∂1 ∂2 u ∂22 u −∂1 ∂2 u ∂12 u

!

,

which is a symmetric matrix. The identifications J0 : Λ0 (K) 7→ C ∞ (R), J1 : Λ1 (V) 7→ C ∞ (M) and J2 : Λ2 (W) 7→ C ∞ (W) are defined by J0 : uχ 7→ u, ! # ! ! " w12 −w11 w12 w11 , dx2 = dx1 + J1 w22 −w21 w22 w21 J2 (ω, µ)dx1 ∧ dx2 = (ω, µ), where we have used the correspondence between a skew-symmetric matrix uχ and a scalar function u. With these vector proxies, the BGG construction yields the complex (.). In 3D, the vector differential forms read         w13 w12 w11 w1          w2  ,  w21  dx1 +  w22  dx2 +  w23  dx3 , w33 w32 w31 w3 16

     w13 w12 w11        w21  dx2 ∧ dx3 +  w22  dx3 ∧ dx1 +  w23  dx1 ∧ dx2 , w33 w32 w31 

with the matrix proxies   w1    w2  , w3



w11   w21 w31

w12 w22 w32

 w13  w23  , w33



w11   w21 w31

w12 w22 w32

Here, a vector can be identified with a skew-symmetric matrix as    0 −w3 w1    w =  w2  ∼ Skw(w) :=  w3 0 −w2 w1 w3

 w1    w2  dx1 ∧ dx2 ∧ dx3 , w3 

 w13  w23  , w33

therefore the skew-symmetric matrix valued forms can be written as

 w1    w2  . w3 

 w2  −w1  , 0

(.)

Skw(w1 , w2 , w3 ), Skw(w11 , w21 , w31 )dx1 + Skw(w12 , w22 , w32 )dx2 + Skw(w13 , w23 , w33 )dx3 , Skw(w11 , w21 , w31 )dx2 ∧ dx3 + Skw(w12 , w22 , w32 )dx3 ∧ dx1 + Skw(w13 , w23 , w33 )dx1 ∧ dx2 , Skw(w1 , w2 , w3 )dx1 ∧ dx2 ∧ dx3 , and the matrix proxies are obvious as discussed above. The identifications J0 : Λ0 (W) 7→ C ∞ (V × K), J1 : Λ1 (K) 7→ C ∞ (M), J2 : Λ2 (V) 7→ C ∞ (M) and J3 : Λ3 (W) 7→ C ∞ (W) are defined by J0 (ω, µ) = (Skw−1 ω, Skw µ), 

 w11 w12 w13   J1 [Skw(w11 , w21 , w31 )dx1 + Skw(w12 , w22 , w32 )dx2 + Skw(w13 , w23 , w33 )dx3 ] =  w21 w22 w23  , w31 w32 w33          w11 w12 w11 w12 w13 w13          J2  w21  dx2 ∧ dx3 +  w22  dx3 ∧ dx1 +  w23  dx1 ∧ dx2  =  w21 w22 w23  , w31 w32 w31 w32 w33 w33 J3 [(ω, µ)dx1 ∧ dx2 ∧ dx3 ] = (ω, µ).

In the vector/matrix form, the S1 operator has the form S1 W = W T − tr(W )I, and

5

1 S1−1 U = U T − tr(U )I. 2

Derivation of the Poincar´ e operators

Next we define the Poincar´e operators for the elasticity complex. 17

Poincar´ e operators on subcomplexes We first give a general result on the construction of Poincar´e operators on subcomplexes. Assume that W i ⊆V i , (W, d) is a subcomplex of (V, d) and the following diagram commutes, i.e. Πi+1 d = dΠi , d ✲ i d ✲ i+1 ✲ V i−1 ✲ ··· V V ··· Πi−1

Πi

Πi+1

(.)

❄ ❄ ❄ d ✲ d✲ ✲ W i−1 ✲ ··· . Wi W i+1 ··· We assume that Πi is surjective for each i, therefore there exists a right inverse of Πi , which we denote as Π†i : W i 7→ V i . Suppose the top row has Poincar´e operators pi : V i 7→ V i−1 satisfying pi+1 d + dpi = id, then the next theorem shows how we can construct the Poincar´e’s operator ˜pi : W i 7→ W i−1 for the bottom row based on pi and the pseudo inverses of the operators Πi . Lemma 1. If Π† commutes with the differential operator d, i.e. dΠ†i = Π†i+1 d,

(.)

the formula ˜pi := Πi−1 pi Π†i defines an operator ˜ pi : W i 7→ W i−1 for the subcomplex (W, d) satisfying di−1 ˜pi + ˜pi+1 di = id. Proof. We have d˜pi = Πi dpi Π†i , and ˜pi+1 d = Πi pi dΠ†i . Therefore d˜pi + ˜pi+1 d = Πi Π†i = id.

If Πj is a projection, the inclusion operator i : W j 7→ V j naturally defines a right inverse of Πj and satisfies the commutative relation (.).

18

Poincar´ e operators on the elasticity complex The construction is summarized in (.). B1

Λ0 (W)

W

C1

Λ0 (W)

F1

Λ0 (W)

(0, µ) l µ

Λ3 (W)

F˜1

(ω, µ) ↓ ω

S ↑ S P1

C ∞ (V)

J2−1 F˜2

C ∞ (M)

M ↓ sym(M )

C ∞ (S)

S ↑ S

P2

M ↓ sym(M ) P3

C ∞ (S)

id

Λ3 (W)

0

Λ3 (W)

0

J3−1 F˜3

C ∞ (M)

0

id F3

Λ2 (V)

J1−1

C ∞ (V × K)

V×K

F2

Λ1 (K)

C3

Γ2

(ω, S1−1 dω) l ω

J0−1

V×K

C2

Γ1

id

W

B3

Λ2 (W)

(0, µ) (ω, µ) (ω, S1−1 dω) (ω, µ) ↑ ↓ ↑ ↓ (ω, S1−1 dω) (ω, S1−1 dω) (0, µ) (0, µ + dS1−1 ω)

id

W

B2

Λ1 (W)

C ∞ (K × V)

0

(0, µ) (ω, µ) ↑ ↓ µ µ − div ω C ∞ (V)

0

(.) The first step is to define an operator in the W-valued de Rham complex Bk : Λk (W) 7→ Λk−1 (W) by Bk =

···

✲ Λk−1 (K)

✲ Λk (K)

−Tk pk

pk 0

!

✲ ···

✛ Tk

Tk

+

+

Tk

2

1

✲ Λk (V)

(.)

d ✲ Λk+1 (K)

✛

✛ ···

.

d d ✲ Λk+1 (V) ✲ Λk+2 (V)

✲ ···

Here Tk : Λk (V) 7→ Λk−1 (K) plays a similar role to S, but with the opposite direction (S has degree 1 while T has degree −1). We define Tk as Tk = pk K − Kpk . 19

By straightforward calculations, we can check that Ak−1 Bk + Bk+1 Ak = idΛk (W) .

(.)

We have derived the homotopy inverses of the first row of (.). The next step is to perform several projections based on Lemma 1. To compute C3 , we use the following path (pω − T µ, pµ) ✛

B3

(ω, µ) ✻ (.)

❄
0, pµ + dS1−1 (pω − T µ)

(ω, µ)


Therefore C3 maps (ω, µ) to 0, pµ + dS1−1 (pω − T µ) . Similarly, we can obtain C2 : B2 (−T µ, pµ) ✛

(0, µ) ✻ (.)

❄
−T µ, −S1−1dT µ

(0, µ)

For C1 , we just have C1 = B1 . Furthermore, for the third row we find F1 , F2 and F3 by:
pω − T S1−1 dω, pS1−1 dω ✛ (ω, S1−1 dω) ✻

(.) ❄ pω −

T S1−1 dω, pS1−1 dω




−T µ, −S1−1dT µ ✛


ω (0, µ) ✻ (.)

❄ −T µ

µ


0, pµ + dS1−1 (pω − T µ) ✛

(ω, µ) ✻ (.)

❄ pµ + dS1−1 (pω − T µ)

(ω, µ)

The next step is to consider vector proxies given by J0 , J1 , J2 and J3 .

20

Matrix proxy We give the vector-matrix forms of the above constructions. Now for u ∈ Λ1 (V), we have Z 1 (1 − t)x ∧ (J1 u)tx · xdt, T1 u(x) ∼ − 0

and so for a matrix ω ∈ M, this gives Z 1 Z 1 Z F˜1 : ω 7→ ω(tx) · xdt + (1 − t)x ∧ (ω(tx) × ∇) · xdt, 0

0

1 0



  1 (ω × ∇)T − (ω × ∇)I · x dt . 2

2

If u ∈ Λ (V), then

1

Z

T2 u(x) ∼

t(1 − t)x ∧ (J2 u)tx ∧ xdt,

0

and so for µ ∈ M, we have F˜2 : µ 7→

Z

1

t(1 − t)x ∧ µtx ∧ xdt = x ∧

0

Z

1 0

 t(1 − t)µtx dt ∧ x.

(.)

Lastly, with u ∈ Λ3 (V), Z

T3 u(x) ∼ −

1

t2 (1 − t)x ∧ (J3 u)tx ⊗ x dt,

0

∞

∞

and so for µ ∈ C (K) × C (M), this leads to   Z 1 Z Z 1 −1 2 2 ˜ t µ ⊗ xdt + S1 F3 : (ω, µ) 7→ t (1 − t)x ∧ µ ⊗ x dt × ∇ + 0

1

t2 [x ⊗ ω − 1/2(x · ω)I] × ∇ dt.

0

0

(.)

Finally, we perform several symmetrizations to get the Poincar´e operators for the elasticity complex. If ω is a symmetric matrix, we have tr(ω × ∇) = 0. Therefore P1 can be interpreted as Z 1 Z 1 (1 − t)x ∧ (∇ × ω(tx)) · x dt, ω ∈ C ∞ (S), ω(tx) · xdt + P1 : ω 7→ 0

0

which is the Ces` aro-Volterra formula. Moreover, the vector proxy of P2 reads: Z 1  Z P2 : µ 7→ symT2 µ = sym t(1 − t)x ∧ µtx ∧ xdt = x ∧ 0

0

1

 t(1 − t)µtx dt ∧ x.

(.)

whenever µ is symmetric. For P3 , we have P3 : µ 7→ sym(pµ + dS1−1 T µ) = sym

Z

0

1

 Z t2 µ ⊗ xdt + S1−1

0

1

  t2 (1 − t)x ∧ µ ⊗ xdt × ∇ , (.)

where S1−1 M := M T − 1/2tr(M ). For any vector u, the matrix x ∧ u ⊗ x has the index form (x ∧ u ⊗ x)il = ǫijk xj uk xl , from which we can easily see that tr (x ∧ u ⊗ x) = ǫijk xj uk xi = 0. Therefore P3 is reduced to Z 1 Z 1   P3 : µ 7→ sym t2 x ⊗ µdt − t2 (1 − t)x ⊗ µ ∧ xdt × ∇ . (.) 0

0

21

2D elasticity complex The construction for the Poincar´e operators for the 2D elasticity complex can be summarized in the diagram below.

Λ0 (W)

W

B1

Λ1 (W)

B2

Λ2 (W)

(0, µ) (ω, µ) (ω, S0−1 dω) (ω, µ) ↑ ↓ ↑ ↓ −1 −1 (0, µ) (ω, S0 dω) (ω, S0 dω) (0, µ + dS0−1 ω) W

Γ0

C1

(ω, S0−1 dω) l ω

W

Λ0 (K)

F1

(0, µ) l µ

Λ1 (V)

J0−1

W

C ∞ (R)

F˜1

C ∞ (R)

C ∞ (M) S ↑ S

P1

F2

Λ2 (W)

0

Λ2 (W)

0

J2−1 F˜2

M ↓ sym(M )

C ∞ (S)

id

id

J1−1

id

W

C2

Γ1

0

P2

C ∞ (W)

0

(0, µ) (ω, µ) ↑ ↓ µ µ − div ω C ∞ (V)

0

(.)

The derivation for the 2D Poincar´e operators is analogous to the 3D case discussed above. The results in Theorem 1 can thus be obtained in a similar manner. We omit the details.

6

Conclusion

In this paper, we proposed an approach to derive null-homotopy operators for the elasticity complex. By construction they automatically satisfy the homotopy relation DP + PD = id. The complex property P 2 = 0 and the polynomial-preserving property were also shown. These properties can naturally be regarded as a gauge condition, since for any u ∈ Vi+1 with Du = 0, the potential φ = Pu ∈ Vi satisfying Pφ = 0 and Dφ = u is uniquely determined. The method discussed in this paper would work for any complex obtained by the BGG construction. The elasticity complex is just one special case, and more examples can be found in, e.g., [5, 20]. Therefore, Poincar´e operators for these complexes can be also constructed following an analogous approach. 22

Acknowledgement The authors are grateful to Ragnar Winther for several useful discussions. The second and third authors were supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339643.

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