Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

Denis, Khimin,; Steinbach, Marc C.; Wick, Thomas

doi:10.1016/j.jcp.2022.111554

Cited by 11 publications

(8 citation statements)

References 55 publications

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“…All numerical computations are performed with the open source software libraries deal.II [5] and DOPELIB [6]. We notice that six other numerical tests were analyzed in detail in [1].…”

Section: Numerical Testsmentioning

confidence: 99%

“…A proof for existence of a global solution and further results of a related quasi-static formulation can be found in [2]. The NLP ( 5) is solved by the reduced approach; see [4] for parabolic problems, which served as starting point of our own algorithms with all details found in [1].…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…with induced norms ∥ • ∥ and ∥ • ∥ I . Moreover we define the restricted inner products [1] (u(t), v(t)) {∂tφ(t)>0} := (u(t), v(t)), ∂ t φ(t) > 0, 0, else, (u, v) {∂tφ>0,I} := I (u(t), v(t)) {∂tφ(t)>0} dt.…”

mentioning

confidence: 99%

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Space‐time phase‐field fracture optimal control computations

Denis

Steinbach

Wick

2023

Proc Appl Math and Mech

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In this work, we undertake additional computational performance studies of a recently developed space-time phase-field fracture optimal control framework. Therein, the phase-field forward problem is formulated in a monolithic fashion. The optimal control problem is formulated with the help of the reduced approach in which the state variable is represented with a solution operator applied to the control. To this end, a Newton algorithm in the control variable is formulated for which auxiliary equations must be solved. Two numerical experiments demonstrate the capabilities of our framework.

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“…All numerical computations are performed with the open source software libraries deal.II [5] and DOPELIB [6]. We notice that six other numerical tests were analyzed in detail in [1].…”

Section: Numerical Testsmentioning

confidence: 99%

Section: Spatial Discretizationmentioning

confidence: 99%

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Space‐time phase‐field fracture optimal control computations

Denis

Steinbach

Wick

2023

Proc Appl Math and Mech

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“…Here, 𝜅 denotes a (bulk) regularization parameter that helps extending the displacements to the entire domain Ω, 𝜀 is a phase-field regularization parameter, 𝛾 is a penalty parameter for the crack irreversibility condition 𝜑 𝑖 ≤ 𝜑 𝑖−1 , 𝜂 denotes a viscosity parameter, and 𝐺 𝑐 is the critical energy release rate. For further explanation on phase-field fracture, and the physical interpretation of the involved parameters, we refer to [48,49,33].…”

Section: Extension To Phase-field Fracturementioning

confidence: 99%

“…and 𝜇 1 = 𝐸 2(1+𝜈) , cf., [33]. The desired phase-field continues the initial notch to the left, i.e., 𝜑 𝑑 (𝑥, 𝑦) 0, 𝑥 ∈ [0.25, 0.5] and 𝑦 ∈ [0.5 − 0.0221, 0.5 + 0.0221] 1, else.…”

Section: Extension To Phase-field Fracturementioning

confidence: 99%

Optimizing Fracture Propagation Using a Phase-Field Approach

Hehl¹,

Mohammadi

Neitzel³

et al. 2021

International Series of Numerical Mathematics

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Within this chapter, we discuss control in the coefficients of an obstacle problem. Utilizing tools from H-convergence, we show existence of optimal solutions. First order necessary optimality conditions are obtained after deriving directional differentiability of the coefficient to solution mapping for the obstacle problem. Further, considering a regularized obstacle problem as a constraint yields a limiting optimality system after proving, strong, convergence of the regularized control and state variables. Numerical examples underline convergence with respect to the regularization. Finally, some numerical experiments highlight the possible extension of the results to coefficient control in phase-field fracture.

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Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

Junker,

Wick

2023

Comput Mech

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We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.

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Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

Cited by 11 publications

References 55 publications

Space‐time phase‐field fracture optimal control computations

Space‐time phase‐field fracture optimal control computations

Optimizing Fracture Propagation Using a Phase-Field Approach

Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

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