Capítulo 2: Metodología de cálculo de los engranajes plásticos
2.7 Validación a través de una ejemplo de cálculo
Proof. Fixv∈ U0. Using the form ofE∗ derived in Lemma 4.1, we obtain
δE∗(uF), v= 1 2 L ξ=−L+1 ρ∈R Vρ(FR)Dρv∗(ξ). (4.19) Now let v(ξ) = Gξ +w(ξ), where G = v(L)/L. Then w∗ is periodic and hence (4.19) implies thatδE∗(uF), w= 0. Hence, we obtain
δE∗(uF), v=δE∗(uF), uG+δE∗(uF), w = 12(2L)
ρ∈R
VρGρ=L∂FW(F)G=∂FW(F)v(L). Inserting this result into the definition ofδErfl, we obtain
δErfl(uF), v=∂FW(F)v(L) +
∞
L ∇vdx= 0. This clearly implies(F).
Further remarks
(1) One often also demands that the QNL interface correction satisfies a local energy consistency condition (see Eet al.(2006) and Ortner and Zhang (2012)):
(E) Local energy consistency. Φiξ(uF) = Φaξ(uF) for all F ∈ R, ξ = K + 1, . . . , L−1.
However, it is straightforward to show (Ortner 2012,§6.4.3) that(F)implies a weaker global version of(E). It is therefore unclear whether the condition
(E) is required at all, except of course to approximate absolute energies. Indeed, we shall not use it in our subsequent 1D analysis.
(2)As remarked above, it is unknown whether the general geometric con- sistency equations have a solution. It is clear from comparing the number of parameters to the number of equations that the corresponding linear op- erator must have a non-trivial kernel. It would therefore be interesting to augment the consistency equations with additional constraints that induce further desirable properties of the resulting a/c coupling.
(3) For 2D pair interactions, Shapeev (2011) has provided a remarkably simple explicit construction of an a/c coupling satisfying(F), which does not use the ideas of geometric reconstruction. Unfortunately the construction cannot be applied to general many-body interactions, but it has proved useful for the analysis of a/c couplings of many-body interactions (Ortner 2012). We will also frequently apply 1D variants of these ideas throughout our consistency analysis. A recent idea applicable to 3D pair interactions is presented in Makridakis, Mitsoudis and Rosakis (2012).
4.6. Force-based a/c coupling
We have so far described two a/c couplings that reduce or remove the ghost force. While the construction of the B-QCE method is straightforward for general a/c interfaces in 2D/3D, the construction of energy-based a/c cou- plings without ghost forces in 2D/3D remains a challenging open problem. An alternative is to construct a/c couplings for the forces, and to accept a non-conservative force field. The simplest method of this type is the force-based quasicontinuum (QCF) method. LetK =L, and define
Fζqcf(uh) := ∂Ea(uh) ∂uh(ζ), ζ ∈ {1, . . . , K}, ∂Ec(u h) ∂uh(ζ), ζ ∈ N ◦ h \ {1, . . . , K}. (4.20) Then we aim to solve the nonlinear system
Fqcf(uh), vh=f, vhh for all vh ∈ Uh, (4.21) where Fqcf(uh), vh= ζ∈N◦ h Fζqcf(uh)vh(ζ).
This scheme is naturally free of ghost forces: Fζqcf(uF) = 0 for all finite element nodesζ ∈ Nh◦.
The first analysis of the QCF method (Dobsonet al.2010b) revealed that while the operator is naturally consistent, its stability is a subtle issue. For example, it was shown that δFqcf(0) is nearly always indefinite and never uniformly stable as an operator from Uh to Uh∗. This not only makes the stability analysis in 2D/3D particularly challenging, but in fact it is cur- rently unknown whether the QCF method is stable in any suitable function space setting.
To overcome this difficulty, Lu and Ming (2013) proposed applying the blending idea behind the B-QCE method to the QCF method. For earlier variants of force-based blending see Fischmeister et al. (1989), Kohlhoff
et al.(1991) and Badiaet al. (2007). To introduce this scheme, let K < L andβ :R→[0,1] with β = 0 in [0, K] andβ = 1 in [L, N], and define
Fζbqcf(uh) := (1−β(ζ))∂E a(u h) ∂uh(ζ) +β(ζ) ∂Ec(u h) ∂uh(ζ) . (4.22) In the blended QCF (B-QCF) method, we aim to solve the nonlinear system
Fζbqcf(uh), vh=f, vhh for all vh ∈ Uh.
Like the QCE and B-QCE methods, the QCF and the B-QCF methods are straightforward to implement, using purely atomistic and purely continuum finite element assembly techniques.
4.7. Formal outline of the error analysis
The fundamental theorem of numerical analysis states, loosely speaking, thatconsistency and stabilityimply convergence. A key motivation of this article is to flesh out this principle in the non-standard setting of a/c cou- pling methods. In the present section, we outline a general framework, in order to motivate the analysis of Sections 5–8.
LetFac∈Ck−1(Uh;Uh∗) be the force operator associated with one of the a/c coupling methods that we constructed in the previous sections; that is, solutions of the a/c coupling under consideration satisfy
Fac(uac
h ), vh=f, vhh for all vh∈ Uh. (4.23) We now turn (4.23) into an error equation. We first assume the existence of an atomistic solution ua to (3.17). Next, we define a quasi-best ap- proximation Πhua∈ Uh (Section 5). Accepting the decay hypothesis (3.20), assuming thatK is sufficiently large and the finite element mesh sufficiently fine, we will obtain that Πhua is ‘close’ to ua. Let eh := uach −Πhua; then (4.23) is equivalent to Fac(Πhua+eh)− Fac(Πhua), vh=δEa(ua)− Fac(Πhua), vh −f, vhZ+− f, vhh (4.24) =:ηacint, vh − ηext, vh,
whereηintac is the consistency error arising from the approximation of the in- ternal energy andηext the consistency error arising from the approximation
of the external forces.
If ηintac = ηext = 0, then a solution to (4.24) is eh = 0. Thus, if ηacint and ηext are sufficiently ‘small’, and if the linearization of the right-hand side of
Table 4.1.
Error Origin Approximation
contribution parameter(s)
far-field error reduction to finite domain N coarsening error finite element discretization h Cauchy–Born replacing the atomistic model L modelling error with continuum model in [L, N]
coupling error interface treatment K, L, β
(4.24) is an isomorphism (we will prove in Section 7 thatδFac(Πhua)v, v ≥ (Cstab)−1∇v2L2), then we expect from the inverse function theorem that
a locally unique solutioneh to (4.24) exists, and that ∇ehL2 ≤CstabηacintU∗ h+ηextUh∗ , (4.25) where ηU∗ h := sup vh∈Uh\{0} η, vh ∇vhL2. (4.26)
We will analyse the consistency errors in Section 6. We will split the error contributions as shown in Table 4.1. While the coarsening error is a standard component of classical finite element error analysis, the combination with the continuum modelling error and in particular the interfacial error requires new ideas. We will obtain consistency error estimates, similar to those in Section 2, depending on the smoothness of ua, the mesh coarsening, the computational domain size, and possibly other approximation parameters.
Next, we will analyse the stability of a/c schemes in Section 7. We shall prove for the B-QCE, B-QCF, and QNL methods that the approx- imation parameters can be chosen in such a way that positivity ofδ2Ea(ua) (see (3.19)) implies positivity ofδFac(Πhua).
Next, we will optimize all approximation parameters to obtain consistency error estimates, purely in terms of the computational cost,
ηintacU∗
h+ηextUh∗ N −β
Th ,
for some exponent β > 0. We will also translate the stability results from Section 7 to this setting, showing that if NTh is sufficiently large then
δFac(Π
hua) is stable.
Finally, in Section 8 we will make the formal argument presented in this subsection fully rigorous. We will obtain results of the form: Let ua be a strongly stable solution of (3.17) satisfying(DH). Then forNTh sufficiently large there exists a solutionuacof (4.23) satisfying∇uac−∇uaL2 NT−β
5. Coarsening error
Our first step in the error analysis is an analysis of the finite element coars- ening error. The results in this section are fairly standard, and hence we only give brief summaries of certain auxiliary results that we require later on. Proofs are given, for the sake of completeness, in the Appendix.
5.1. Best approximation operator
Recall the definition of the nodal interpolation operatorIhfrom Section 4.2. Since Ih does not mapU toUh, and to avoid any error contributions from the atomistic region (and possibly a neighbourhood of the interface region), we define
Πhu(x) :=
Ihu(x), x∈[0, L+rcut],
Ihu(x)− Nx−−LL−−rrcutcutu(N), x∈[L+rcut, N].
With this definition, Πhu∈ Uh for all u:Nh→R.
In our coarsening error analysis below, it will be useful to split the inter- polant into Πhu = Ihu+ (Πhu−Ihu). Hence, we separately estimate the interpolation errors as follows.