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 -