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Mixed finite element methods for linear elasticity with weakly imposed symmetry
Douglas N. Arnold, Richard S. Falk, Ragnar Winther
TL;DR
Simple stable mixed finite elements for elasticity remain difficult, especially when stress symmetry is imposed strongly. The paper weakly imposes symmetry through an extended Hellinger–Reissner formulation and derives discrete elasticity complexes from de Rham complexes, obtaining stable spaces including piecewise linear stresses and piecewise constant displacements.
Problem
Stable simple mixed finite element spaces for elasticity have remained difficult to construct, while direct mixed approximations are important for elasticity and related models.
Method
The paper uses an extended Hellinger–Reissner formulation with a skew-symmetric multiplier and constructs discrete elasticity complexes systematically from de Rham discretizations.
Results
The authors construct stable finite element spaces satisfying the required stability conditions, including piecewise linear stress and piecewise constant displacement approximations.
Takeaways & Limitations
The de Rham-based construction provides a systematic route to simpler weakly symmetric mixed elasticity elements without stabilizing bubbles or multiplier continuity.
Takeaways & Limitations
Strongly symmetric discrete complexes require high-order polynomial stresses, including piecewise quartic stresses with 162 degrees of freedom per tetrahedron in three dimensions.
Abstract
from arXiv · showhide
In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.
1. Introduction
The paper addresses the difficulty of constructing simple stable mixed finite elements for elasticity by weakly imposing stress symmetry and deriving the elasticity complex from the de Rham complex.
- Motivation: Stable mixed elasticity elements have remained difficult to construct, with earlier low-order approaches requiring piecewise cubic stresses and many degrees of freedom.In two dimensions, one cited lowest-order stress space uses piecewise cubics with 24 degrees of freedom per triangle.
- Weakly imposed symmetry: The extended Hellinger–Reissner formulation introduces a skew-symmetric Lagrange multiplier to enforce stress symmetry weakly.The formulation is posed over H(div, Ω; M) × L2(Ω; V) × L2(Ω; K), where K is the space of skew-symmetric matrices.
- Weakly imposed symmetry: The extended and strongly symmetric continuous formulations are equivalent, but their discrete solutions generally need not correspond.This distinction creates additional possibilities for finite element discretization while limiting direct equivalence of the discrete systems.
- Paper contribution: The authors construct the elasticity complex from the de Rham complex and mimic that construction discretely to derive simpler mixed finite elements systematically.The resulting stable discretizations include piecewise linear stress and piecewise constant displacement approximations.
- Broader motivation: Mixed formulations also remain relevant for models with nonlocal stress–strain relations and are uniformly stable in the incompressible limit.For viscoelasticity and plasticity, stress elimination may be impossible; mixed methods remain an alternative formulation.
2. Notation, statement of main results, and preliminaries
The paper formulates weakly symmetric mixed elasticity discretizations using finite element spaces linked by commuting discrete complexes. Its main result identifies stable polynomial spaces derived from known de Rham discretizations.
- Discrete formulation: The weakly symmetric schemes approximate stress, displacement, and a skew-symmetric multiplier in Σh, Vh, and Qh.These spaces satisfy Σh ⊂ H(div, Ω; M), Vh ⊂ L2(Ω; V), and Qh ⊂ L2(Ω; K).
- Stability: Stability of the saddle-point system requires coercivity on the constrained stress space and an inf-sup condition for displacement–multiplier pairs.The constants in these conditions are required to be independent of h.
- Main result: For r ≥ 0, Nédélec second-family H(div) elements of degree r + 1 for stress and discontinuous degree-r polynomials for displacement and multiplier yield stable spaces.The construction requires neither stabilizing bubble functions nor interelement continuity for the multiplier.
- Motivation: The approach is motivated by the successful construction of stable mixed elements for Poisson problems through de Rham-complex discretizations.An analogous commuting diagram for the elasticity complex guides the elasticity construction.
- Complex-based construction: The construction uses commuting projections between continuous and discrete complexes, so exactness of the continuous complex implies exactness of the discrete one.The discrete operators are projections determined by the finite element degrees of freedom.
3. The elasticity complex
The elasticity complex organizes the differential operators underlying elasticity formulations, with a weakly imposed stress symmetry producing an alternative complex for discretization. Directly discretizing the strongly symmetric complex leads to high-order stress spaces, motivating the weak-symmetry approach.
- Strongly imposed symmetry: The elasticity complex characterizes the differential structure associated with strongly imposed stress symmetry.It is exact on contractible domains, and its construction involves operators such as curl, J, and divergence.
- Discretization challenge: Pointwise symmetry makes commuting discretizations require high-order polynomial stresses, including piecewise quartics with 162 degrees of freedom per tetrahedron in three dimensions.The corresponding two-dimensional construction used piecewise cubic stresses.
- Weakly imposed symmetry: The weak-symmetry formulation replaces the symmetric-stress complex with a matrix-valued complex that permits additional discretization possibilities.The extended formulation introduces matrix fields and a skew-symmetric variable while remaining equivalent to the original elasticity system.
- Relationship between complexes: The weak-symmetry elasticity complex is obtained from the strongly symmetric complex through a projection relationship.The paper presents this connection as a route to relating the two complexes.
- Relationship between complexes: Both the strongly and weakly symmetric elasticity complexes are ultimately derived from the de Rham complex.The paper states that the weak-symmetry complex follows from the de Rham complex and that the strongly symmetric one follows as well.
4. From the de Rham to the elasticity complex
The paper constructs the weak-symmetry elasticity complex as an exact subcomplex derived from a transformed de Rham complex. This provides the basis for constructing stable discrete elasticity complexes and mixed finite elements.
- Construction: The construction begins with a W-valued de Rham complex, where W combines skew-symmetric matrices and vectors.The transformed complex is built using an isomorphism involving the algebraic operator K.
- Exactness result: When the de Rham complex is exact, the elasticity complex is exact on contractible domains.This is stated as Theorem 4.1.
- Construction: The transformed differential operator has the form A(ω, µ) = (dω − Sµ, dµ), with S = dK − Kd.The operator S is purely algebraic and independent of the spatial point.
- Exact subcomplex: A constrained subcomplex is formed by spaces Γ1 and Γ2, and the transformed operator maps successive spaces into one another.The resulting diagram commutes, so exactness transfers from the de Rham complex to the subcomplex.
- Identification with elasticity: Under natural identifications, the differential-form operators correspond to row-wise gradient, inclusion of skew fields, J, row-wise divergence, and the skew operator.This identifies the constructed subcomplex with the weakly symmetric elasticity complex.
5. The discrete construction
The discrete construction derives an elasticity complex from two compatible discrete de Rham complexes, using commuting projections and a surjectivity condition. Exactness of the resulting discrete sequence supports stable finite element spaces.
- Construction: Two discretizations of the de Rham sequence are combined under a compatibility condition to construct a discrete elasticity complex.The construction defines discrete analogues of the continuous operators and inserts discrete isomorphisms into the de Rham sequences.
- Construction: Surjectivity of S1,h is required because the discrete operator need not be invertible in the applications.Under this assumption, S1,h has a right inverse, enabling discrete projection operators and a commuting diagram.
- Exactness: Commuting projections make the constructed bottom row a subcomplex of the top row, so exactness transfers from the top complex to the discrete one.The theorem states that exactness of both discrete de Rham sequences implies exactness of the discrete elasticity sequence.
- Stability: The resulting exact discrete elasticity sequence suggests finite element spaces that lead to stable discretizations.The paper postpones specific space choices and later verifies stability for selected discrete de Rham complexes.
- Stability: Commutativity of the relevant diagram provides a sufficient condition for S1,h to be surjective.This condition is used to establish the key assumption for the selected spaces.
6. A family of discrete elasticity complexes
The paper instantiates the discrete construction with polynomial differential-form spaces indexed by r ≥ 0. In the lowest-order case, the resulting method uses piecewise linear stresses and piecewise constant displacements and multipliers.
- Polynomial spaces: The examples form a family of discrete elasticity complexes indexed by the polynomial degree r ≥ 0.Specific discrete de Rham complexes are chosen first, and the resulting complexes are then used to derive elasticity finite elements.
- Lowest-order case: The lowest-order method uses piecewise linear functions for stresses and piecewise constant functions for displacements and multipliers.This is the paper’s explicit low-order approximation choice.
- Polynomial spaces: The construction assumes a contractible polyhedral domain triangulated by tetrahedra and uses the two principal families of piecewise polynomial differential-form spaces.These include spaces associated with the standard H(curl) and H(div) discretizations.
- Polynomial spaces: The spaces for 0-forms correspond to Lagrange subspaces of H1, while the spaces for 3-forms correspond to discontinuous piecewise polynomials in L2.For intermediate form degrees, the spaces correspond to Nédélec discretizations of H(curl) and H(div).
- Stability: The selected spaces give discrete de Rham sequences with commuting projections, and the key surjectivity assumption is verified through diagram commutativity.The construction then concludes that S1,h is surjective for these choices.
7. Stable mixed finite elements for elasticity
The paper constructs stable weak-symmetry mixed finite element spaces from discrete elasticity complexes derived from de Rham discretizations. The resulting family uses Nédélec second-family stress spaces and discontinuous polynomial spaces for displacement and the multiplier, with quasi-optimal convergence estimates.
- Construction: The spaces Σh, Vh, and Qh are obtained from appropriate matrix- and vector-valued spaces in discrete de Rham sequences.This construction transfers the elasticity-complex framework into finite element spaces systematically.
- Finite element spaces: For r ≥ 0, Nédélec second-family H(div) elements of degree r + 1 approximate stress, while discontinuous degree-r polynomials approximate displacement and the multiplier.The corresponding spaces are Σh ∼ Pr+1Λ2(Th; V), Vh ∼ PrΛ3(Th; V), and Qh ∼ PrΛ3(Th; K).
- Stability: Stability follows from div Σh ⊂ Vh and a stronger surjectivity property for the combined displacement–multiplier space.The proof establishes the discrete stability conditions (A1) and (A2), including a bounded construction for the stronger condition (A2′).
- Stability: The projection analysis establishes the boundedness estimates needed to prove the discrete stability conditions, with constants independent of meshsize.Some projection operators require modified definitions because edge degrees of freedom are not bounded on H1.
- Convergence: The stable spaces yield convergence and error estimates, including a quasi-optimal bound for stress, displacement, and multiplier approximation.Theorem 7.2 bounds the combined error by the best approximation error in Σh, Vh, and Qh, with a constant independent of h.
- Limitation: Because multiplier approximation is one order less accurate than stress approximation, the extra variable prevents improved estimates analogous to those for Poisson’s equation.The limitation concerns the approximation contribution of p to the displacement and stress error estimates.
8. A simplified element
The paper simplifies the stable element by reducing selected stress-space degrees of freedom while preserving the complex and required surjectivity. The lowest-order simplified construction retains stability and uses piecewise linear stress with piecewise constant displacement and multiplier spaces.
- Lowest-order element: At lowest order, the simplified stable element uses piecewise linear stress and piecewise constant displacement and multiplier spaces.The analogue of Theorem 7.2 holds for the simplified spaces with r = 0.
- Design: The simplified element leaves Vh and Qh unchanged while reducing tangential–normal stress components on each tetrahedral face to a reduced linear space.The goal is a stable element that is simpler than the full-linear construction.
- Design: The construction reduces face degrees of freedom while retaining P1Λ1(T; V) and setting unused face degrees of freedom to zero.The reduced space is built by separating the standard projection component from the remaining face-based component.
- Degrees of freedom: The reduced 1-form space has 48 degrees of freedom: 36 edge degrees of freedom and 12 face degrees of freedom.Its unisolvency follows from the unisolvency of the constituent spaces.
- Degrees of freedom: The corresponding reduced 2-form stress space has 24 face degrees of freedom, fewer than the 36 degrees of freedom of P1Λ2(T; V).The reduced degrees of freedom are selected through a six-dimensional face space on each face.
- Stability: The reduced spaces form an exact complex and preserve the required surjectivity, including the map onto P0Λ3(Th; V).These properties are inherited through the construction from the exact complex and the retained subspaces.