PSI - Issue 74

Jiří Vala et al. / Procedia Structural Integrity 74 (2025) 91–98

95

J.Vala & V.Koza´k / Structural Integrity Procedia 00 (2025) 000–000

5

An evident alternative form of (5) is ( γ, C − 1 ( τ m

1 τ m

s /β ) = ( γ,ε ( u m

m s − 1

m s − 1

) /α ) + h ( γ, C −

s − u

)) .

(6)

s − τ

Clearly u m m = 1 of simple functions on I , whose convergence to the solution of (3) and (2) must be verified. The detailed explanation of all step of this proof exceeds the limited extent of this conference paper. However, most their ideas can be analogous to those of Vala and Toma´sˇ (2025) where n = 1 is considered for simplicity (except brief remarks to potential generalizations), thus Λ is always empty, and all proofs are performed in two levels: • with D = 0 everywhere (which generates a linear problem), and • with a more realistic D , whose evaluation is compatible with the above presented 5-step algorithm. Moreover, we need to implement • the traction separation phenomena on Λ , which can be inspired by a straightforward modification of Vala and Koza´k (2020). The overview of steps of this proof, covering such three levels, is: 1. Existence of solution of (4) and (5) with fixed s , m : For n = 1 and zero-valued D this can be guaranteed by the standard Lax-Milgram theorem, see Roub´ıcˇek (2005), Part 2.3; the validity of its assumptions comes from the Sobolev embedding and trace theorems, supported by the Cauchy-Schwarz inequality, and from the Korn inequality. In all other cases, fortunately, to cover the second and third left-hand-side additive terms in (4), another classical existence result is available in Bre´zis (1968). However, the exploitation of properties of a regularizing kernel K ∈ L 2 ( Ω × Ω ), occurring in the third step of the evaluation of D ( . ), by Dra´bek and Milota (2013), Part 2.2, is necessary. 2. Uniform a priori bounds for { u } ∞ m = 1 in L 2 ( I , V ) and { τ } ∞ m = 1 in L 2 ( I , E ): For fixed s , m it is possible to choose v = u m s − u m s − 1 in (4) and γ = τ m s in (6). Rather extensive calculations lead to the estimate ∥ τ m r ∥ 2 H + ∥ u m s ∥ 2 V ≤ c    1 + r  s = 1 ∥ u m s − 1 ∥ 2 V    . (7) for any r ∈ { 1 , . . . , m } , independent of the choice of m , and a positive constant c . Then the discrete Gronwall lemma, see Roub´ıcˇek (2005), Part 1.6, yields this result. 3. Weak limits of { u } ∞ m = 1 in L 2 ( I , V ) and { τ } ∞ m = 1 in L 2 ( I , E ): Their existence, up to subsequences, comes from the Eberlein-Shmul’yan theorem, following Bu¨hler and Salomon (2017), Part 3.4. Notice that stronger results rely on compact Sobolev embeddings. 4. Limit passage from (4) and (5) to (3) and (2): The exploitation of the above sketched convergence properties is su ffi cient. 4. Computational issues An evident advantage of the above sketched approach is its independency of the discretization of Ω , Θ , Γ and Λ . However, all resulting equations for particular time steps are still in infinite-dimensional function spaces, thus such a discretisation is needed for practical calculations. This can be done in the form ( ε ( v δ ) ,τ m s ) + ( ε ( v δ ) , (1 −D ( u m δ s ∗ )) C ε ( u m δ s )) + ⟨ [ v δ ] , T ( u m δ s ∗ ) ⟩ ∗ = ( v δ , f m s ) + ⟨ v δ , g m s ⟩ (8) for all v δ ∈V δ and ( γ δ , C − 1 ( τ m δ s − τ m δ s − 1 ) /α ) + h ( γ δ , C − 1 τ m δ s /β ) = ( γ δ ,ε ( u m δ s − u m δ s − 1 )) . (9) for all γ δ ∈G δ ,with δ understood as some reference mesh element edge length. G δ and V δ in In (8) and (9) are finite dimensional approximation of G and V (not their subspaces necessarily). Namely u m δ s ∗ are certain approximations of u m δ s ∗ for, step by step, s ∈ { 1 , . . . , m } ; e. g. their choice as u m δ s − 1 can be seen as a starting point of an appropriate iterative procedure. Such numerical methods as the inexact Newton method, the nonlinear conjugate gradient method, of the Nelder-Mead simplex method are available; the preference of just one approach is impossible because of di ff erent techniques of evaluation of D ( . ) and T ( . ), including numerous upgrades of the so-called Mazars model for concrete by Giry et al. (2011), Havla´sek et al. (2016) and Arruda et al. (2023). Namely the standard finite element techniques manifest poor results in the evaluation of T ( . ). One can trace three decades of the history of extended and generalized finite element methods and similar algorithms, paying attention s and τ m s from (4) and (5) (if they exist) generate certain sequences { u m } ∞ m = 1 and { τ m } ∞