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Lie-group interpolation and variational recovery for internal variables

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Abstract

We propose a variational procedure for the recovery of internal variables, in effect extending them from integration points to the entire domain. The objective is to perform the recovery with minimum error and at the same time guarantee that the internal variables remain in their admissible spaces. The minimization of the error is achieved by a three-field finite element formulation. The fields in the formulation are the deformation mapping, the target or mapped internal variables and a Lagrange multiplier that enforces the equality between the source and target internal variables. This formulation leads to an \(L_2\) projection that minimizes the distance between the source and target internal variables as measured in the \(L_2\) norm of the internal variable space. To ensure that the target internal variables remain in their original space, their interpolation is performed by recourse to Lie groups, which allows for direct polynomial interpolation of the corresponding Lie algebras by means of the logarithmic map. Once the Lie algebras are interpolated, the mapped variables are recovered by the exponential map, thus guaranteeing that they remain in the appropriate space.

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Acknowledgments

Support for this work was received through the U.S. Department of Energy’s (DOE) Advanced Simulation and Computing (ASC) Program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. This work was also supported in part by the Joint U.S. Department of Defense (DoD)/DOE Munitions Technology Development Program.

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Authors and Affiliations

  1. Mechanics of Materials Department, Sandia National Laboratories, Livermore, CA, 94550, USA

    Alejandro Mota, WaiChing Sun, Jakob T. Ostien & James W. Foulk III

  2. Solid Mechanics Department, Sandia National Laboratories, Albuquerque, NM, 87185, USA

    Kevin N. Long

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  1. Alejandro Mota
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  2. WaiChing Sun
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  3. Jakob T. Ostien
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Correspondence to Alejandro Mota.

Appendix: error minimization

Appendix: error minimization

Proposition

The projection defined by (2.10) is orthogonal with respect to \(V_h\) and therefore the distance between source and target fields is minimal in the \(L_2\) norm of \(V\).

Proof

Define the inner product in \(V\) as

$$\begin{aligned} ({\varvec{u}}, {\varvec{v}}) := \int \limits _B {\varvec{u}}\cdot {\varvec{v}}\ dV, \quad \forall \, {\varvec{u}}, {\varvec{v}}\in V, \end{aligned}$$
(9.1)

and the corresponding \(L_2\) norm as

$$\begin{aligned} ||{\varvec{u}}|| := \left( \int \limits _B {\varvec{u}}\cdot {\varvec{u}}\ dV \right) ^{\frac{1}{2}} \ge 0, \quad \forall \, {\varvec{u}}\in V. \end{aligned}$$
(9.2)

We follow the approach outlined by Brenner and Scott [4]. The variational statement (2.5) may be written after discretization as

$$\begin{aligned} \int \limits _B (\bar{{\varvec{z}}}_h - {\varvec{z}}) \cdot {\varvec{\zeta }}_h \ dV = 0 \end{aligned}$$
(9.3)

where \({\varvec{z}}\in V\) and \(\bar{{\varvec{z}}}_h, {\varvec{\zeta }}_h \in V_h\). Using the inner product notation introduced in (9.1) this discrete statement takes the form

$$\begin{aligned} (\bar{{\varvec{z}}}_h - {\varvec{z}}, {\varvec{\zeta }}_h) = 0, \quad \forall {\varvec{\zeta }}_h \in V_h \end{aligned}$$
(9.4)

which shows the fundamental orthogonality relation between the projection error \(\bar{{\varvec{z}}}_h - {\varvec{z}}\) and the space \(V_h\), therefore proving that (2.10) is an orthogonal projection. Introduce the Cauchy-Schwarz inequality

$$\begin{aligned} ({\varvec{u}}, {\varvec{v}}) \le ||{\varvec{u}}|| ||{\varvec{v}}||, \quad \forall {\varvec{u}}, {\varvec{v}}\in V, \end{aligned}$$
(9.5)

then for any \({\varvec{u}}_h \in V_h\)

$$\begin{aligned} ||\bar{{\varvec{z}}}_h \!-\! {\varvec{z}}||^2&= (\bar{{\varvec{z}}}_h \!-\! {\varvec{z}}, \bar{{\varvec{z}}}_h \!-\! {\varvec{z}}) \nonumber \\&= (\bar{{\varvec{z}}}_h \!-\! {\varvec{z}}, {\varvec{u}}_h \!-\! {\varvec{z}}) \!+\! (\bar{{\varvec{z}}}_h \!-\! {\varvec{z}}, \bar{{\varvec{z}}}_h \!-\! {\varvec{u}}_h)\nonumber \\&= (\bar{{\varvec{z}}}_h \!-\! {\varvec{z}}, {\varvec{u}}_h \!-\! {\varvec{z}}) \quad \text{ from } (9.4) \ \text{ with } \ {\varvec{\zeta }}_h \!=\! \bar{{\varvec{z}}}_h - {\varvec{u}}_h\nonumber \\&\le ||\bar{{\varvec{z}}}_h \!-\! {\varvec{z}}|| ||{\varvec{u}}_h \!-\! {\varvec{z}}|| \quad \text{ from } \ (9.5). \end{aligned}$$
(9.6)

If \(||\bar{{\varvec{z}}}_h - {\varvec{z}}|| = 0\) then (9.6) is satisfied trivially and (2.10) is optimal. If \(||\bar{{\varvec{z}}}_h - {\varvec{z}}|| > 0\) then it follows that \(||\bar{{\varvec{z}}}_h - {\varvec{z}}|| \le ||{\varvec{u}}_h - {\varvec{z}}||\). As \({\varvec{u}}_h\) is any element in \(V_h\), this inequality is also satisfied when taking the infimum

$$\begin{aligned} ||\bar{{\varvec{z}}}_h - {\varvec{z}}|| \le \inf \{||{\varvec{u}}_h - {\varvec{z}}|| : {\varvec{u}}_h \in V_h \}, \end{aligned}$$
(9.7)

and by the definition of the infimum,

$$\begin{aligned} \inf \{||{\varvec{u}}_h - {\varvec{z}}|| : {\varvec{u}}_h \in V_h \} \le ||\bar{{\varvec{z}}}_h - {\varvec{z}}||, \end{aligned}$$
(9.8)

therefore to satisfy both (9.7) and (9.8) it follows that

$$\begin{aligned} ||\bar{{\varvec{z}}}_h - {\varvec{z}}|| = \inf \{||{\varvec{u}}_h - {\varvec{z}}|| : {\varvec{u}}_h \in V_h \}. \end{aligned}$$
(9.9)

The infimum exists for some \({\varvec{u}}_h \in V_h\), thus (9.9) is actually a minimum

$$\begin{aligned} ||\bar{{\varvec{z}}}_h - {\varvec{z}}|| = \min \{||{\varvec{u}}_h - {\varvec{z}}|| : {\varvec{u}}_h \in V_h \}, \end{aligned}$$
(9.10)

which shows that the projection (2.10) is optimal.

An alternate approach to determine that (2.10) is optimal is to introduce the difference or error between \(\bar{{\varvec{z}}}_h\) and \({\varvec{z}}\) into a variational principle as

$$\begin{aligned} \Pi [\bar{{\varvec{z}}}_h] := ||\bar{{\varvec{z}}}_h - {\varvec{z}}||^2 = \int \limits _B (\bar{{\varvec{z}}}_h - {\varvec{z}}) \cdot (\bar{{\varvec{z}}}_h - {\varvec{z}}) \ dV. \end{aligned}$$
(9.11)

Introducing as before the test function \({\varvec{\zeta }}_h\), and upon minimization by applying variations, this leads to

$$\begin{aligned} D\Pi [\bar{{\varvec{z}}}_h]({\varvec{\zeta }}_h) = 2 \int \limits _B (\bar{{\varvec{z}}}_h - {\varvec{z}}) \cdot {\varvec{\zeta }}_h \ dV = 0, \end{aligned}$$
(9.12)

which is equivalent to (9.3) and which implies that (2.10) is optimal as it minimizes the error.

These two methods show that the variational principle (2.2) results in a target field \(\bar{{\varvec{z}}}_h\) that minimizes the error in the norm (9.2) with respect to the source field of internal variables \({\varvec{z}}\). Note that the error (9.11) is zero if and only if the source field \({\varvec{z}}\) is already a member of the discrete space \(V_h\). Thus, in general \(\bar{{\varvec{z}}}_h\) is not equal to \({\varvec{z}}\), and in the evaluation of the equilibrium condition (2.6) with the target internal variables in a new mesh, there are two options: satisfy equilibrium by changing the deformation mapping and maintaining the target internal variables constant, or satisfy equilibrium by evolving the target internal variables and maintaining the deformation mapping constant.

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Mota, A., Sun, W., Ostien, J.T. et al. Lie-group interpolation and variational recovery for internal variables. Comput Mech 52, 1281–1299 (2013). https://doi.org/10.1007/s00466-013-0876-1

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