A Mixed Finite Element Method for Coupled Plates

Organization of the Paper.

Section 2 revisits the plane elasticity model and the Kirchhoff plate model, presenting the variational formulation for two coupled plates based on displacements and providing its well-posedness. Section 3 introduces the mixed variational formulation and constructs the nonstandard Sobolev space Σ for two coupled plates. The well-posedness of the mixed formulation is proven. Section 4 presents the conforming mixed finite element method and an illustrative example. Numerical experiments are performed to validate the theory.

2.1 Hypotheses and Notations

Let H m ⁢ ( D ; X ) denote the Sobolev space of functions of L 2 ⁢ ( D ; X ) within domain D, taking values in space X, whose distributional derivatives up to the order m also belong to L 2 ⁢ ( D ; X ) . Let C m ⁢ ( D ; X ) denote the space of m-times continuously differentiable functions and P l ⁢ ( D ; X ) denote the set of all the polynomials with the total degree no greater than l. In this paper, X could be ℝ , ℝ 2 or 𝕊 , where 𝕊 denotes the set of all symmetric ℝ 2 × 2 matrices. If X = ℝ , then H m ⁢ ( D ) abbreviates H m ⁢ ( D ; X ) , similarly for C m ⁢ ( D ) and P l ⁢ ( D ) . The standard Sobolev norm ∥ ∙ ∥ m , D will be taken. When m = 0 , H 0 ⁢ ( D ; X ) is exactly L 2 ⁢ ( D ; X ) . The L 2 inner product is denoted by ( ∙ , ∙ ) D for the scalar, vector-valued, and tensor-valued L 2 spaces over D. In particular, for the tensor-valued L 2 space,

For scalar functions v, vector-valued functions 𝝍 , and matrix-valued functions 𝝉 , the first-order differential operators ∇ ⁡ v , ∇ ⁡ 𝝍 , ∇ s ⁡ 𝝍 , div ⁡ 𝝍 and Div ⁡ 𝝉 are defined by

Define the following spaces:

Let S be the midsurface of a thin plate which lies in the x-y plane. For simplicity, assume that S is clamped on ∂ ⁡ S 0 ⊂ ∂ ⁡ S , with nonzero measure, and is free on the complementary part which is denoted as ∂ ⁡ S 1 , i.e., ∂ ⁡ S = ∂ ⁡ S 0 ∪ ∂ ⁡ S 1 , ∂ ⁡ S 0 ∩ ∂ ⁡ S 1 = ∅ . Other boundary conditions can be treated similarly. In this paper, the deformations are assumed to be small and governed by the equations of plane elasticity, and the material is supposed to be homogeneous and isotropic. The load in S is assumed to be a distributed load ( 𝒇 , f 3 ) ≔ ( f 1 , f 2 , f 3 ) , which can be decomposed into in-plane and out-of-plane components. Let 𝒏 and 𝒕 denote the unit outer normal vector and the unit counterclockwise tangential vector of ∂ ⁡ S , respectively, and let ∂ n ⁡ ( ∙ ) ≔ ∇ ⁡ ( ∙ ) ⋅ 𝒏 and ∂ t ⁡ ( ∙ ) ≔ ∇ ⁡ ( ∙ ) ⋅ 𝒕 .

For the in-plane part, the plane elasticity model is

where 𝒖 = ( u 1 , u 2 ) T is the displacement vector in plane, and 𝝈 is the stress tensor. The constitutive relation reads

where 𝐈 is the identity tensor and tr ⁡ ( ∙ ) is the trace operator. Here t , E , and ν denote the thickness of the plate, Young’s modulus, and the Poisson ratio of the material, respectively.

For the out-of-plane part, the Kirchhoff plate model is

where u 3 is the deflection and 𝑴 is the moment tensor. The constitutive relation reads

The components of 𝑴 and the shear force T on ∂ ⁡ S are denoted by

The set 𝒱 ∂ ⁡ S 1 contains all corner points x where exist two adjacent edges e 1 , e 2 ⊂ ∂ ⁡ S 1 satisfying e 1 ∩ e 2 = x . These edges do not meet an angle of π and possess different normal and tangential vectors 𝒏 1 , 𝒕 1 and 𝒏 2 , 𝒕 2 , respectively. For x ∈ 𝒱 ∂ ⁡ S 1 with x being the end point of e 1 and the starting point of e 2 , define

2.2 Two Coupled Plates Model

This subsection presents the model of two coupled plates with midsurfaces denoted by S and ~ ⁢ S . Throughout this paper, let ( ~ ∙ ) denote the quantities related to ~ ⁢ S with the same physical meaning as those in S without exception. For simplicity, assume that the material parameters t , E , ν and ~ ⁢ t , ~ ⁢ E , ~ ⁢ ν take the same value, respectively.

Suppose that S and ~ ⁢ S are coupled along a common rectilinear junction Γ ≔ ∂ ⁡ S ∩ ∂ ⁡ ~ ⁢ S , see Figure 1. Assume that S is clamped on ∂ ⁡ S 0 ⊂ ∂ ⁡ S with nonzero measure. Let ∂ ⁡ S 1 and ∂ ⁡ ~ ⁢ S 1 be the free boundary of S and ~ ⁢ S , respectively. Then it holds that

Let 𝒍 = 𝒏 × 𝒕 . It follows that ( 𝒏 , 𝒕 , 𝒍 ) defined on Γ ⊂ ∂ ⁡ S constitutes a local orthogonal reference system of S. By taking the opposite direction ∼ 𝒕 = - 𝒕 on Γ, the local orthogonal reference system of ~ ⁢ S can be defined by ( ~ 𝒏 , ∼ 𝒕 , ~ 𝒍 ) on Γ ⊂ ∂ ⁡ ~ ⁢ S . Define the angle θ between two plates by cos ⁡ θ = 𝒏 ⋅ ~ ⁢ 𝒏 . Then

When the plates are coplanar, the angle is equal to π, and when the plates coincide, the angle is equal to 0. This paper deals with the case 0 < θ < π .

Assume that two plates are coupled with a rigid junction. Then the equilibrium equations of two coupled plates, see [8, 42] for instance, are

(2.4)

with the boundary conditions

(2.5)

and the junction conditions

(2.6)

Here ∂ ⁡ Γ denotes the collection of two end points of Γ.

2.3 Variational Formulation Based on Displacements

This subsection recalls the variational formulation based on displacements from [8, 42]. Let ϕ ≔ ( 𝒖 , u 3 ; ~ ⁢ 𝒖 , ~ ⁢ u 3 ) and F ≔ ( 𝒇 , f 3 ; ~ ⁢ 𝒇 , ~ ⁢ f 3 ) denote the displacement and external distribute force of S ∪ ~ ⁢ S , respectively. Define

(2.7)

equipped with the norm

Given ϕ , ψ ∈ W , define

Introduce the L 2 space

equipped with the norm

and the corresponding L 2 inner product ( ∙ , ∙ ) V . Then the variational formulation for two coupled plates (2.4)–(2.6e) with an angle θ reads as follows.

3.1 Spaces and Operators

By introducing the union of the stresses and moments

in (2.4) as an independent variable, this section provides a mixed formulation for two coupled plates. Note that, for a single plate S, the mixed formulation based on H ⁢ ( div , 𝕊 ) × L 2 ⁢ ( ℝ 2 ) is considered for plane elasticity and the mixed formulation based on H ⁢ ( divDiv , 𝕊 ) × L 2 is considered for the Kirchhoff plate. Readers can refer to [25, 5, 2, 14, 21, 26, 27, 24, 20, 1, 3, 6] and [23, 44, 11, 13, 17, 12, 22] for these two formulations with corresponding mixed finite element methods, respectively. The mixed formulation for two coupled plates can be regarded as an extension of mixed formulations for a single plate.

The generalization of mixed formulations for a single plate to two coupled plates is not straightforward. The main challenge lies in providing an appropriate Sobolev space that incorporates both free boundary conditions (2.5b)–(2.5c) and junction conditions (2.6c)–(2.6e), which are defined on parts of the boundary for stresses and moments. For example, given a stress tensor 𝝈 ∈ H ⁢ ( div , S ; 𝕊 ) , the restriction of 𝝈 ⁢ 𝒏 on ∂ ⁡ S belongs to H - 1 / 2 ⁢ ( ∂ ⁡ S ) . Since H - 1 / 2 ⁢ ( ∂ ⁡ S ) is defined as a dual space, 𝝈 ⁢ 𝒏 cannot be simply restricted on ∂ ⁡ S 1 [10, Section 2.5]. Similar considerations apply to functions in H ⁢ ( divDiv , 𝕊 ) [18]. Furthermore, recall the continuity conditions (2.6d) on the junction as

These conditions involve the coupling of stresses and moments, which need to deal with the coupling of traces of H ⁢ ( div , 𝕊 ) and H ⁢ ( divDiv , 𝕊 ) .

In this paper, instead of directly tackling the free boundary and rigid junction conditions, a nonstandard Sobolev space is proposed through the application of duality arguments. This approach avoids the direct use of trace operators in nonstandard Sobolev spaces, which would be technically rather involved. Similar results can be found in [39, 38, 37] to deal with H - 1 ⁢ ( divDiv , 𝕊 ) . Introduce the L 2 space

equipped with the norm

and the corresponding L 2 inner product ( ∙ , ∙ ) V s . Define the operator

on W in (2.7). Note that the first and the second differential operators are defined in the context of classical weak derivatives. It can be observed that B ⁢ W ⊂ V s .

Let X be a Hilbert space such that W is dense in X, which will be specified later. Introduce the operator B * : D ⁢ ( B * ) ⊂ V s → X * , where D ⁢ ( B * ) is the domain of definition of B * , as follows: y ∈ D ⁢ ( B * ) if and only if y ∈ V s and there exists a linear functional G ∈ X * such that

Define B * ⁢ y = G . Note that 〈 B * ⁢ y , x 〉 is well-defined for x ∈ X and y ∈ D ⁢ ( B * ) , and

The domain D ⁢ ( B * ) is a Hilbert space with the graph norm

Note that the displacement formulation in Problem 1 can be viewed as the case X = W . This paper considers the case X = V with V from (2.9). Obviously, W ⊂ V and W is dense in V. The operator B is then a densely defined linear operator. It follows from (2.10) that V * = V and 〈 ∙ , ∙ 〉 V * × V = ( ∙ , ∙ ) V . Define a subspace of V s as follows:

According to the definition of B * , Φ ∈ D ⁢ ( B * ) if and only if Φ ∈ V s and there exists a linear functional G ∈ V such that

Note that C 0 ∞ ⁢ ( S ; ℝ 2 ) × C 0 ∞ ⁢ ( S ) × C 0 ∞ ⁢ ( ~ ⁢ S ; ℝ 2 ) × C 0 ∞ ⁢ ( ~ ⁢ S ) is contained in W and is dense in V. This, (3.3) and (3.4) show that Φ ∈ H s for any Φ ∈ D ⁢ ( B * ) and

Let Σ ≔ D ⁢ ( B * ) be the space for the introduced independent variable. Then the following equality holds:

The nonstandard Sobolev space Σ can be explicitly given by

equipped with the norm ∥ Φ ∥ Σ ≔ ( ∥ Φ ∥ V s 2 + ∥ B * ⁢ Φ ∥ V 2 ) 1 / 2 , i.e.,

3.2 Mixed Formulation and Its Well-Posedness

The results in the previous subsection of densely defined operator B and its adjoint B * together with their definition of domain motivate the mixed formulation as follows. Recall the definition of 𝒞 1 in (2.1) and 𝒞 2 in (2.2). Define 𝒞 ≔ ( 𝒞 1 , 𝒞 2 ; 𝒞 1 , 𝒞 2 ) by

It is easy to verify that ( Φ , Ψ ) 𝒞 - 1 is a symmetric, nonnegative bilinear form. Brezzi’s theory (see, e.g., [10]) shows that the well-posedness of Problem 2 holds if the following conditions are satisfied:

1. ( Φ , Ψ ) 𝒞 - 1 is bounded: There exists a constant a > 0 such that

   | ( Φ , Ψ ) 𝒞 - 1 | ≤ a ⁢ ∥ Φ ∥ Σ ⁢ ∥ Ψ ∥ Σ   for all  ⁢ Φ , Ψ ∈ Σ .

2. ( B * ⁢ Ψ , ψ ) V is bounded: There exists a constant b > 0 such that

   | ( B * ⁢ Ψ , ψ ) V | ≤ b ⁢ ∥ Ψ ∥ Σ ⁢ ∥ ψ ∥ V   for all  ⁢ Ψ ∈ Σ , ψ ∈ V .

3. ( Ψ , Ψ ) 𝒞 - 1 is coercive on the kernel of B: There exists a constant α > 0 such that

   ( Ψ , Ψ ) 𝒞 - 1 ≥ α ⁢ ∥ Ψ ∥ Σ 2   for all  ⁢ Ψ ∈ Ker ⁡ B ,

   with

   Ker ⁡ B = { Ψ ∈ Σ : ( B * ⁢ Ψ , ψ ) V = 0 ⁢  for all  ⁢ ψ ∈ V } .

4. ( B * ⁢ Ψ , ψ ) V satisfies the inf-sup condition: There exists a constant β > 0 such that

   inf 0 ≠ ψ ∈ V ⁡ sup 0 ≠ Ψ ∈ Σ ⁡ ( B * ⁢ Ψ , ψ ) V ∥ Ψ ∥ Σ ⁢ ∥ ψ ∥ V ≥ β .

These conditions are known as Brezzi’s conditions with constants a , b , α , and β.

The following theorems follow directly from the construction of the space Σ. For completeness, the detailed deductions are provided here. The theorem below shows that the solution of Problem 1 is also a solution of Problem 2, which can be utilized to prove the inf-sup condition of Problem 2.