H^2- Korn's Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model

04/17/2021 ∙ by Hongliang Li, et al. ∙ Chinese Academy of Science 0

We establish a new H2 Korn's inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle [41] and the NZT tetrahedron [45] are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct new regularized interpolation estimate and the enriching operator for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results are consistent with the theoretical prediction.

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1. Introduction

Let be the solution of the following boundary value problem:

(1.1)

where and are the Lamé constants, and is the microscopic parameter satisfying . In particular, we are interested in the regime when is close to zero. This boundary value problem arises from a linear strain gradient elastic model proposed by Aifantis et al [4, 37], and is the displacement. This model may be regarded as a simplification of the more general strain gradient elasticity models [32] because it contains only one extra material parameter besides the Lamé constants and . This simplified strain gradient model successfully eliminated the strain singularity of the brittle crack tip field [19], and we refer to [18] and [21] for other strain gradient models.

The boundary value problem (1.1) is essentially a singularly perturbed elliptic system of fourth order due to the appearance of the strain gradient . C-conforming finite elements such as Argyris triangle [5] seems a natural choice for discretization. The performance of Argyris triangle and several other C-conforming finite elements has been carefully studied in [20] for a nonlinear strain gradient elastic model. A drawback of the C

-conforming elements is that the number of the degrees of freedom is large and high order polynomial has to be used in the shape functions, which is more pronounced for three dimensional problems; See, e.g., the finite element for a three-dimensional strain gradient model proposed in 

[36] locally has degrees of freedom. We aim to develop some simple and robust nonconforming elements to avoid such difficulties. The robustness is understood in the sense that the elements converge uniformly in the energy norm with respect to .

To this end, we firstly prove a new HKorn’s inequality and its discrete analog in any dimension. This HKorn’s inequality may be viewed as a quantitative version of the so-called vector version of J.L. Lions lemma [16, Theorem 6.19-1]; See also (2.12), while our proof is constructive and may be adapted to prove a Korn’s inequality for piecewise Hvector field (broken HKorn’s inequality for short), which may be viewed as a higher-order counterpart of Brenner’s seminal Korn’s inequality [13] for piecewise H vector fields. Compared to the broken Hinequality proved in [27]

, the jump term associated with the gradient tensor of the piecewise vector field may be dropped. Therefore, the degrees of freedom associated with the gradient tensor along each face or edge may be dropped, which simplify the construction of the elements. Based on this observation, all H

conforming but H nonconforming elements are suitable for approximating this strain gradient model. We choose the Specht triangle [41] and the NZT tetrahedron [45] as two typical representatives. The Specht triangle is simpler than those in [27], because the elements therein locally belong to a dimensional subspace of quintic polynomials, while the tensor products of the Specht triangle locally belong to an dimensional subspace of quartic polynomials. It is worth mentioning that the broken HKorn’s inequality may also be exploited to develop C interior penalty method [17, 11] for the strain gradient elastic model.

To prove the robustness of both elements, we construct a regularized interpolation operator and an enriching operator, and derive certain estimates for such operators, which are key to prove sharp error estimate for problems with less smooth solution. These two operators are also useful for strain gradient elasticity model with other type boundary conditions.

The remaining part of the paper is organized as follows. We prove the continuous and the broken HKorn’s inequalities in §2. The Specht triangle and the NTZ tetrahedron are introduced in §3 and the corresponding regularized interpolant are constructed and analyzed therein. We introduce enriching operators for both elements in §4, and derive the error bounds uniformly with respect to in the same part. The numerical tests of both elements are reported in the last Section, which confirm the theoretical prediction in §4.

Throughout this paper, the constant may differ from line to line, while it is independent of the mesh size and the materials parameter .

2. HKorn’s Inequalities

In this part we prove the HKorn’s inequalities and the broken HKorn’s inequalities. Let us fix some notations firstly.

2.1. Notations

Let be a bounded convex polytope. We shall use the standard notations for Sobolev spaces, norms and semi-norms [2]. The function space consists functions that are square integrable over , which is equipped with norm and the inner product . Let be the Sobolev space of square integrable functions whose weak derivatives up to order are also square integrable, the corresponding norm with the semi-norm .

For a positive number that is not an integer, is the fractional order Sobolev space. Let be the largest integer less than and . The sem-inorm and the norm are given by

By [2, §7], the above definition for the fractional order Sobolev space is equivalent to the one obtained by interpolation, i.e.,

In particular, there exists that depends on and such that

(2.1)

For is the closure in of the space of functions with compact supports in . In particular,

where is normal derivative of .

For any vector-valued function , its gradient is a matrix-valued function given by for . The strain tensor is given by with . The divergence operator is defined by . The spaces and are standard Sobolev spaces for the vector fields. Without abuse of notation, we employ to denote the abstract value of a scalar, the norm of a vector, and the Euclidean norm of a matrix. Throughout this paper, we may drop the subscript whenever no confusion occurs.

Let be a simplicial triangulation of with maximum mesh size . We assume all elements in are shape-regular in the sense of Ciarlet and Raviart [15], i.e., there exists a constant such that , where is the diameter of element , and is the diameter of the largest ball inscribed into , and is the so-called chunkiness parameter  [12]. We denote by , and the sets of dimensional faces, edges and vertices, respectively. Let be the set of boundary faces. We denote by the set of interior faces. Similar notations apply to and . We denote by the set of boundary simplexes. Given a simplex or sub-simplex , we let be the element star containing . Similar notations apply to the set of faces, edges and vertices. For example, for any node , the is the set of boundary edges sharing a common node .

We classify the boundary vertices as follows. We say that a node

is a flat node if the boundary edges set span a -dimensional linear space. Otherwise, we say that is a sharp node. We let , where and denote the sets of the flat node and sharp node, respectively. For any sharp node , we may choose a set with linear independent boundary edges, i.e., , such that provide a basis of .

2.2. H-Korn’s inequality

We write the boundary value problem (1.1) into the following variational problem: Find such that

(2.2)

where the bilinear form is defined for any as

and the fourth-order tensors and the sixth-order tensor are defined by

respectively. Here is the Kronecker delta function. The strain gradient is a third order tensor defined by .

The wellposedness of Problem (2.2) depends on the coercivity of the bilinear form over , which is a direct consequence of the following HKorn’s inequality:

(2.3)

This inequality was proved in [27, Theorem 1] by exploiting the community property of the strain operator and the partial derivative operator , which has been implicitly used in [1] to prove an inequality similar to (2.3).

In what follows, we prove that (2.3) remains valid for a more general vector field in . The precise form is stated in (2.4). Our proof relies on the fact that the strain gradient field fully controls the Hessian of the displacement algebraically; See, cf. (2.5). This fact will be further exploited to prove a discrete analog of (2.4) for a piecewise vector field, which is dubbed as the broken HKorn’s inequality. We have exploited a weaker version of such broken HKorn’s inequality to design two robust strain gradient finite elements in [27].

Theorem 2.1.

For any and , there holds and

(2.4)

Here, the constant in right hand side of the above inequality may be replaced by when .

Proof.

The core of the proof is the following algebraic inequality:

(2.5)

Integrating (2.5) over domain , we obtain

(2.6)

which together with the first Korn’s inequality [25, 26]:

(2.7)

implies (2.4).

To prove (2.5), we start with the identity

(2.8)

where vanishes for .

Employing the elementary algebraic inequality

we obtain

(2.9)

Using the elementary algebraic equality

we obtain

(2.10)

Combining  (2.8),  (2.9) and  (2.10), we obtain (2.5) immediately.

vanishes when . Therefore, the constant in the right-hand side of (2.4) may be replaced by . This completes the proof. ∎

A direct consequence of Theorem 2.1 is the following full HKorn’s inequality.

Corollary 2.2.

Let be a domain such that the following Korn’s inequality is valid for any vector field and ,

If and , then and

(2.11)

In [16, Theorem 6.19-1], the following vector version of J.L. Lions Lemma is proved: For any domain in and , then

(2.12)

The inequality (2.11) may be viewed as a quantitative version of (2.12) with , while the proof in [16, Theorem 6.19-1] is nonconstructive and is not easy to be adapted to prove the broken HKorn’s inequality for a piecewise vector filed.

The regularity of Problem (2.2) is essential to prove a uniform error estimate. Unfortunately, it does not seem easy to identify such estimates in the literature, and we give a proof for the readers’ convenience. We firstly make an extra regularity assumption.

Hypothesis 2.3.

Let be a solution of following equations,

where . Then there holds that for any ,

(2.13)

If is a smooth domain, the regularity property (2.13) is standard; See e.g., [3]. While it is unclear whether the above regularity estimate is true for a convex polytope. Nevertheless, if the operator is replaced by the Laplacian operator, then (2.13) is proved in [31, Chapter 4 Theorem 4.3.10].

Lemma 2.4.

Assume Hypothesis 2.3 is valid and let be the solution of (2.2), then there exists that may depend on but independent of such that

(2.14)

where satisfies

(2.15)

Moreover, we have

(2.16)

and

(2.17)

Under Hypothesis 2.3, we may prove this regularity result by following essentially the same line of the proof in [35, Lemma 5.1]. We include it here for completeness.

Proof.

By (1.1) and (2.15), we have

Using the regularity hypothesis  (2.13), we obtain

(2.18)

By the standard regularity estimate, we have

(2.19)

Denoting and integration by parts, we have

where .

Using the regularity estimate (2.19), we obtain

Using the trace inequality (2.24), we obtain, for to be chosen later,

Using (2.6), we obtain

Using the regularity estimates (2.18) and (2.19), we bound the right-hand side of the above inequality as

Combining the above inequalities, we obtain

Choosing properly, we obtain (2.14).

Using (2.14) and Poincaré inequality, and noting that , we obtain

(2.20)

and

(2.21)

Interpolating (2.20) and (2.14) with , we obtain

Using the interpolation inequality (2.1), we obtain

A combination of the above two inequalities yields (2.16).

Combining (2.18) and (2.14), we obtain

Interpolating the above inequality and (2.21), we obtain (2.17). ∎

2.3. The broken H-Korn’s inequality

For any , the space of piecewise vector fields is defined by

which is equipped with the broken norm

where with . Moreover, . For any , we denote by the jump of across the faces or the edge.

The main result of this part is the following broken H-Korn’s inequality.

Theorem 2.5.

For any , there exits that depends on and but independent of such that

(2.22)

where is the projection and

where is the tangential vector of the face (or edge for ), and is the infinitesimal rigid motion on .

For a piecewise vector filed , the inequality (2.22) improves the one proved in [27, Theorem 2] by removing the jump term

This term stands for the jump of the gradient tensor of the vector field across the element boundary. This would simplify the construction of the robust strain gradient elements as shown in the next two parts.

Proof of Theorem 2.5 Integrating (2.5) over element , we obtain,

Summing up all , we get

(2.23)

which together with the following Korn’s inequality for a piecewise H vector filed proved by Mardal and Winther [30]

implies (2.22). ∎

We shall frequently use the following trace inequalities.

Lemma 2.6.

For any Lipschitz domain , there exists depending on such that

(2.24)

For an element , there exists independent of , but depends on such that

(2.25)

If , then there exists independent of , but depends on and such that

(2.26)

The multiplicative type trace inequality (2.24) may be found in [23, Theorem 1.5.1.10], while (2.25) is a direct consequence of (2.24). The third trace inequality is a combination of (2.25) and the inverse inequality for any polynomial .

3. Interpolation for nonsmooth data

Motivated by the broken HKorn’s inequality (2.22), we conclude that the Hconforming but Hnonconforming finite elements are natural choices for approximating Problem (3.2). A family of rectangular elements in this vein may be found in [29], and two nonconforming tetrahedron elements were constructed and analyzed in [44]. Note that the tensor product of certain finite elements for the singular perturbation problem of fourth order may also be used to approximate (1.1), we refer to [39, 40, 24, 35, 42] and references therein for such elements. In what follows, we select the Specht triangle [41] and the NZT tetrahedron [45] as the representatives. The Specht triangle is a successful plate bending element, which passes all the patch tests and performs excellently, and is one of the best thin plate triangles with degrees of freedom that currently available [48, Quatation in p. 345]. The NZT tetrahedron may be regarded as a three-dimensional extension of the Specht triangle.

The Specht triangle and the NZT tetrahedron may be defined by the finite element triple  [15] in a unifying way as following: is a simplex, and

with extra constraints

(3.1)

where is a dimensional simplex opposite to vertex , and is the edge vector from to . Here is the Zienkiewicz space defined by

where is the barycentric coordinates associated with the vertex .

The finite element space is define by

The corresponding homogenous finite element space is defined by

It is clear that . We denote , and approximating problem reads as: Find such that

(3.2)

where the bilinear form is defined for any as

with

The energy norm is defined as . The bilinear form is coercive in this energy norm as shown in the next lemma.

Lemma 3.1.

For any ,

(3.3)

where appears in the Poincaré inequality

(3.4)

The estimate (3.3) immediately implies the wellposedness of Problem (3.2) for any fixed .

Proof.

For any , there holds

Using the first Korn’s inequality (2.7) and the estimate (2.23), we obtain

The coercivity estimate (3.3) follows by using the Poincare’s inequality (3.4). ∎

The standard interpolation estimate for the above elements reads as [14],

This interpolant is unbounded in , which is even not well-defined for a function in .

Our definition for the regularized interpolant is a combination of a regularized interpolant in [24] and an enriching operator defined in [34].

Define with the Scott-Zhang interpolant [38], where is the quadratic Lagrangian finite element space with vanishing trace. The operator is locally defined as follows.

  1. If is an interior vertex, then we fix an element from ,

  2. If is a flat node, then we fix an element from ,

  3. If is a sharp node, then