Details
| Originalsprache | Englisch |
|---|---|
| Qualifikation | Doctor rerum naturalium |
| Gradverleihende Hochschule | |
| Betreut von |
|
| Datum der Verleihung des Grades | 18 Okt. 2024 |
| Erscheinungsort | Hannover |
| Publikationsstatus | Veröffentlicht - 12 Nov. 2024 |
Abstract
greiche theoretische Grundlage ermöglicht es uns vielseitige numerische Studien für Optimalsteuerungsprobleme mit Phasenfeld-Riss-Ausbreitung durchzuführen. Diese Probleme sind hochgradig nichtlinear und unendlichdimensional, was ihren Lösungsprozess sehr heikel macht. Aus mechanischer Sicht besteht das Hauptziel darin, eine optimale Steuerung zu
bestimmen, die den sich ausbreitenden Riss, repräsentiert durch das Phasenfeld, in eine gewünschte Richtung oder sogar auf einen gewünschten Pfad lenkt. Dies wird durch eine Tracking Kostenfunktion realisiert, die mit einen zusätzlichen Tichonow-Term regularisiert wird. Die Beschränkungen ergeben sich aus einem variationalen Phasenfelmodell, bei dem die Steuerung als eine Neumann Randkraft wirkt. Aus mathematischer Sicht ist das Ziel, diese Probleme in einem stetigen Raum-Zeit-Setting zu formulieren, und effizient zu lösen. Dazu werden drei Formulierungen für das Optimalsteuerungsproblem vorgestellt, die sich vor allem in den Beschränkungen unterscheiden, aber auch in der Art und Weise, wie die Irreversibilität des Risses behandelt wird. In der ersten Formulierung sind die Beschränkungen durch Euler-Lagrange-Gleichungen gegeben, die sich aus einer Minimierung der Rissenergie ergeben. Dabei wird die Irreversibilität des Risses, welche formal einer Ungleichung für die Zeitableitung des Phasenfelds entspricht, durch einen Strafterm ersetzt. Dies führt zu einem gleichungsbeschränkten Optimalsteuerungsproblem, welches durch den reduzierten Ansatz gelöst wird, bei dem die Zustandsvariable mithilfe der Euler-Lagrange-Gleichungen eliminiert wird. Resultierend daraus, müssen vier Hilfsgleichungen gelöst werden, und für jede wird ein eigenes Zeitschrittschema hergeleitet. Hierbei wird ein diskontinuierlicher Galerkin Ansatz in der Zeit und ein kontinuierlicher Galerkin Ansatz im Raum verwendet. Diese theoretischen Herleitungen für Optimalsteuerungsprobleme mit Phasenfeld-Riss-Ausbreitung sind völlig neuartig. Die zweite Formulierung beinhaltet einen modifizierten und insbesondere Fréchet differenzierbaren Strafterm, der zu einer zeitlichen Ableitung der Ordnung zwei für das Phasenfeld führt. Daher werden die entsprechenden Euler-Lagrange-Gleichungen als ein System erster Ordnung in der Zeit reformuliert. In der letzten Formulierung werden. sowohl das untere Phasenfeld-NLP als auch das obere Optimalsteuerungs-NLP als abstrakte NLPs in Banachräumen behandelt. Das Optimalsteuerungs-NLP wird durch die Bedingungen erster Ordnung für das Phasenfeld-NLP beschränkt, genauer durch das zugehörige KKT System. Auf diese Weise wird hier ein Komplementaritätssystem für Optimalsteuerungsprobleme mit Phasenfeld-Riss-Ausbreitung hergeleitet. Dabei wird zunächst die Fréchet Differenzierbarkeit aller auftretenden Kostenfunktionale und Beschränkungen gezeigt. Anschließend wird sowohl die Regularität für das Phasenfeld-NLP als auch das Optimalsteuerungs-NLP unter bestimmten Bedingungen bewiesen. Es werden Optimalitätsbedingungen erster Ordnung für beide Probleme sowie Optimalitätsbedingungen höherer Ordnung für das Phasenfeld-NLP untersucht.
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Hannover, 2024. 150 S.
Publikation: Qualifikations-/Studienabschlussarbeit › Dissertation
}
TY - BOOK
T1 - Space-time modeling, analysis, and finite element simulations of phase-field fracture optimal control problems
AU - Khimin, Denis
PY - 2024/11/12
Y1 - 2024/11/12
N2 - In this dissertation, we introduce for the first time, space-time formulations for phase-field fracture optimal control problems. We develop the necessary theoretical framework, which includes a detailed selection of function spaces, derivation of auxiliary equations for the reduced approach, their Galerkin discretization, and the formulation of Karush-Kuhn-Tucker (KKT) systems. This comprehensive theoretical foundation enables us to establish a space-time framework and conduct extensive computational performance studies for phase-field fracture optimal control problems. These problems are highly nonlinear and infinite-dimensional making their solution process very delicate. From a mechanical point of view, the main objective is to determine an optimal control that steers the propagating fracture, represented by the phase-field, into a desired direction or even a desired path. This is accomplished through a tracking-type cost functional, which is defined as the distance between the current state and the desired state and regularized with an additional Tikhonov term. The constraints are given by a variational phase-field fracture model, where the control acts as a Neumann type boundary force. Mathematically, the ambition extends to reformulating these problems within a continuous space-time framework, thereby examining the efficiency of the solution process. In total, we present three formulations for optimal control problems governed by phase-field fracture that differ mainly in the form of the constraints but also in the way the crack irreversibility condition is handled. In the first formulation, the constraints are given by Euler-Lagrange equations corresponding to a regularized crack energy that shall be minimized (lower-level problem). Therein, the crack irreversibility, which is formally an inequality constraint for the time derivative of the phase-field, is addressed through a penalty approach. This yields an equality-constrained upper-level Nonlinear Optimization Problem (NLP), that is solved by the reduced approach, involving the elimination of the state variable through the Euler-Lagrange equations. As a result, four auxiliary equations must be solved: state, adjoint, tangent, and adjoint Hessian. For each auxiliary problem a time-stepping scheme is derived using discontinuous Galerkin (dG) methods in time and continuous Galerkin (cG) methods in space. This theoretical derivation for space-time optimal control problems with phase-field is novel. The second formulation involves a modified and Fr\'echet differentiable penalty term leading to a second order time derivative for the phase-field. Hence, the corresponding Euler-Lagrange equations are reformulated as a first-order-in-time system. The resulting optimal control NLP is solved again with the reduced approach which includes once more the derivation of the required auxiliary equations. In the final formulation, both the lower-level phase-field NLP and the upper-level optimal control NLP are treated as abstract NLPs in Banach spaces. The optimal control NLP is then constrained by the first order optimality conditions, i.e., KKT system, corresponding to the lower-level NLP. Following this, we establish a rigorous mathematical framework, demonstrating Fr\'echet differentiability for all occurring cost functionals and constraints. Afterwards, we derive first-order optimality conditions for the phase-field NLP and the optimal control NLP . Alongside a discussion on the sufficient optimality conditions for the forward problem, we prove regularity results (constraint qualification) for both, the lower-level and the upper-level NLP under certain conditions.
AB - In this dissertation, we introduce for the first time, space-time formulations for phase-field fracture optimal control problems. We develop the necessary theoretical framework, which includes a detailed selection of function spaces, derivation of auxiliary equations for the reduced approach, their Galerkin discretization, and the formulation of Karush-Kuhn-Tucker (KKT) systems. This comprehensive theoretical foundation enables us to establish a space-time framework and conduct extensive computational performance studies for phase-field fracture optimal control problems. These problems are highly nonlinear and infinite-dimensional making their solution process very delicate. From a mechanical point of view, the main objective is to determine an optimal control that steers the propagating fracture, represented by the phase-field, into a desired direction or even a desired path. This is accomplished through a tracking-type cost functional, which is defined as the distance between the current state and the desired state and regularized with an additional Tikhonov term. The constraints are given by a variational phase-field fracture model, where the control acts as a Neumann type boundary force. Mathematically, the ambition extends to reformulating these problems within a continuous space-time framework, thereby examining the efficiency of the solution process. In total, we present three formulations for optimal control problems governed by phase-field fracture that differ mainly in the form of the constraints but also in the way the crack irreversibility condition is handled. In the first formulation, the constraints are given by Euler-Lagrange equations corresponding to a regularized crack energy that shall be minimized (lower-level problem). Therein, the crack irreversibility, which is formally an inequality constraint for the time derivative of the phase-field, is addressed through a penalty approach. This yields an equality-constrained upper-level Nonlinear Optimization Problem (NLP), that is solved by the reduced approach, involving the elimination of the state variable through the Euler-Lagrange equations. As a result, four auxiliary equations must be solved: state, adjoint, tangent, and adjoint Hessian. For each auxiliary problem a time-stepping scheme is derived using discontinuous Galerkin (dG) methods in time and continuous Galerkin (cG) methods in space. This theoretical derivation for space-time optimal control problems with phase-field is novel. The second formulation involves a modified and Fr\'echet differentiable penalty term leading to a second order time derivative for the phase-field. Hence, the corresponding Euler-Lagrange equations are reformulated as a first-order-in-time system. The resulting optimal control NLP is solved again with the reduced approach which includes once more the derivation of the required auxiliary equations. In the final formulation, both the lower-level phase-field NLP and the upper-level optimal control NLP are treated as abstract NLPs in Banach spaces. The optimal control NLP is then constrained by the first order optimality conditions, i.e., KKT system, corresponding to the lower-level NLP. Following this, we establish a rigorous mathematical framework, demonstrating Fr\'echet differentiability for all occurring cost functionals and constraints. Afterwards, we derive first-order optimality conditions for the phase-field NLP and the optimal control NLP . Alongside a discussion on the sufficient optimality conditions for the forward problem, we prove regularity results (constraint qualification) for both, the lower-level and the upper-level NLP under certain conditions.
U2 - 10.15488/18120
DO - 10.15488/18120
M3 - Doctoral thesis
CY - Hannover
ER -