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Optimization solvers

Module for solving the optimization problem to determine the load factor in crack propagation simulations.

This module provides the LoadFactorSolver class, which implements the GMERR (Generalized Maximum Energy Release Rate) criterion for crack propagation. The solver is based on the work of Amestoy and Leblond (1992) and is designed to work with the gcrack.lefm module. It uses gradient descent with line search to find the optimal crack propagation angle and load factor.

The module also includes a custom gradient descent optimizer with line search for robust convergence, even in the presence of discontinuities or "cups" in critical energy release rate Gc.

References

Amestoy, M., & Leblond, J. B. (1992). A new numerical method for crack growth prediction. International Journal of Solids and Structures, 29(21), 2619-2638. https://doi.org/10.1016/0020-7683(92)90210-K

LoadFactorSolver

Solver for load factors and crack propagation angles using the GMERR criterion.

This class implements the GMERR (Generalized Maximum Energy Release Rate) criterion for crack propagation. It uses automatic differentiation to compute gradients and Hessians of the objective function, and employs gradient descent with line search to find the optimal crack propagation angle and load factor.

Attributes:

Name Type Description
model ElasticModel

The elastic model used for the simulation.
Gc Callable

The critical energy release rate function.
grad Callable

The gradient of the objective function.
hess Callable

The Hessian of the objective function.
grad_pert Callable

The gradient of the perturbed objective function.
hess_pert Callable

The Hessian of the perturbed objective function.
Source code in src/gcrack/optimization_solvers.py 
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class LoadFactorSolver:
    """Solver for load factors and crack propagation angles using the GMERR criterion.

    This class implements the GMERR (Generalized Maximum Energy Release Rate) criterion for crack propagation.
    It uses automatic differentiation to compute gradients and Hessians of the objective function, and employs gradient descent with line search to find the optimal crack propagation angle and load factor.

    Attributes:
        model (gcrack.models.ElasticModel): The elastic model used for the simulation.
        Gc (Callable): The critical energy release rate function.
        grad (Callable): The gradient of the objective function.
        hess (Callable): The Hessian of the objective function.
        grad_pert (Callable): The gradient of the perturbed objective function.
        hess_pert (Callable): The Hessian of the perturbed objective function.
    """

    def __init__(self, model: ElasticModel, Gc_func: Callable, xc: List):
        """Initializes the LoadFactorSolver.

        Args:
            model (gcrack.models.ElasticModel): The elastic model used for the simulation.
            Gc_func (Callable): The critical energy release rate function.
            xc (List): Position of the crack tip.
        """
        # Store the model
        self.model = model
        # Store the crack tip position
        self.xc = xc
        # Set the critical energy release rate function
        self.Gc = jit(Gc_func)
        # Automatic differentiation of the objective function
        self.grad = jit(grad(self.objective))
        self.hess = jit(hessian(self.objective))
        # Automatic differentiation of the perturbed objective function
        self.grad_pert = jit(grad(self.objective_pert))
        self.hess_pert = jit(hessian(self.objective_pert))

    def objective(
        self,
        x: jnp.ndarray,
        Ep: float,
        s: float,
        KIc: float,
        KIIc: float,
        Tc: float,
        KIp: float,
        KIIp: float,
        Tp: float,
        phi0: float,
    ) -> float:
        """Computes the objective function for the GMERR criterion.

        This function computes the objective function for the GMERR criterion, which is used to find the optimal crack propagation angle and load factor.

        Args:
            x (jnp.ndarray): The optimization variables (crack propagation angle).
            Ep (float): Plane strain modulus.
            s (float): Internal length associated with T-stress.
            KIc (float): Mode I stress intensity factor for the controlled problem.
            KIIc (float): Mode II stress intensity factor for the controlled problem.
            Tc (float): T-stress for the controlled problem.
            KIp (float): Mode I stress intensity factor for the prescribed problem.
            KIIp (float): Mode II stress intensity factor for the prescribed problem.
            Tp (float): T-stress for the prescribed problem.
            phi0 (float): Initial crack angle.

        Returns:
            float: The value of the objective function.
        """
        # NOTE: The KIc (etc.) means controlled (not critical!)
        phi = x[0]
        # Compute the G star
        Gs_cc = G_star(phi, phi0, KIc, KIIc, Tc, Ep, s)
        Gs_cp = G_star_coupled(phi, phi0, KIc, KIIc, Tc, KIp, KIIp, Tp, Ep, s)
        Gs_pp = G_star(phi, phi0, KIp, KIIp, Tp, Ep, s)
        # Compute the Gc from phi
        gc = self.Gc(phi)
        # Compute and return the load factor
        delta = Gs_cp**2 - 4 * Gs_cc * (Gs_pp - gc)
        return (-Gs_cp + jnp.sqrt(delta)) / (2 * Gs_cc)

    def objective_pert(
        self,
        x: jnp.ndarray,
        Ep: float,
        s: float,
        KIc: float,
        KIIc: float,
        Tc: float,
        KIp: float,
        KIIp: float,
        Tp: float,
        phi0: float,
    ) -> float:
        """Computes the perturbed objective function for the GMERR criterion.

        This function adds a small perturbation to the objective function to avoid
        convergence to a maximum instead of a minimum.

        Args:
            x (jnp.ndarray): The optimization variables (crack propagation angle).
            Ep (float): Plane strain modulus.
            s (float): Internal length associated with T-stress.
            KIc (float): Mode I stress intensity factor for the controlled problem.
            KIIc (float): Mode II stress intensity factor for the controlled problem.
            Tc (float): T-stress for the controlled problem.
            KIp (float): Mode I stress intensity factor for the prescribed problem.
            KIIp (float): Mode II stress intensity factor for the prescribed problem.
            Tp (float): T-stress for the prescribed problem.
            phi0 (float): Initial crack angle.

        Returns:
            float: The value of the perturbed objective function.
        """
        return (
            self.objective(x, Ep, s, KIc, KIIc, Tc, KIp, KIIp, Tp, phi0) + 1e-5 * x[0]
        )

    def solve(
        self, phi0: float, SIFs_controlled: dict, SIFs_prescribed: dict, s: float
    ):
        """Solves for the optimal crack propagation angle and load factor.

        This function uses gradient descent with line search to find the optimal
        crack propagation angle and load factor.

        Args:
            phi0 (float): Initial crack angle.
            SIFs_controlled (dict): Stress intensity factors for the controlled problem.
            SIFs_prescribed (dict): Stress intensity factors for the prescribed problem.
            s (float): Internal length associated with T-stress.

        Returns:
            Tuple[float, float]: The optimal crack propagation angle and load factor.
        """
        KIc, KIIc, Tc = (
            SIFs_controlled["KI"],
            SIFs_controlled["KII"],
            SIFs_controlled["T"],
        )
        KIp, KIIp, Tp = (
            SIFs_prescribed["KI"],
            SIFs_prescribed["KII"],
            SIFs_prescribed["T"],
        )

        # Perform the minimization
        kwargs = {
            "Ep": self.model.Ep_func(self.xc),
            "s": s,
            "KIc": KIc,
            "KIIc": KIIc,
            "Tc": Tc,
            "KIp": KIp,
            "KIIp": KIIp,
            "Tp": Tp,
            "phi0": phi0,
        }

        phi = gradient_descent_with_line_search(phi0, self.grad, kwargs=kwargs)

        # Check the stability of the solution (i.e., check if solution is a max)
        hess = self.hess([phi], **kwargs)[0][0]
        solution_is_max = hess < 0
        if solution_is_max:
            print("Found a maximum instead of minimum -> perturbating the objective")
            print("Note: this test might also be triggered by cups!")
            # Perform another gradient descent on the perturbed objective
            phi = gradient_descent_with_line_search(phi0, self.grad_pert, kwargs=kwargs)

        # Compute the load factor
        load_factor = self.objective([phi], **kwargs)

        return float(phi), float(load_factor)

__init__(model, Gc_func, xc)

Initializes the LoadFactorSolver.

Parameters:

Name Type Description Default
model ElasticModel

The elastic model used for the simulation.

 required
Gc_func Callable

The critical energy release rate function.

 required
xc List

Position of the crack tip.

 required
Source code in src/gcrack/optimization_solvers.py 
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def __init__(self, model: ElasticModel, Gc_func: Callable, xc: List):
    """Initializes the LoadFactorSolver.

    Args:
        model (gcrack.models.ElasticModel): The elastic model used for the simulation.
        Gc_func (Callable): The critical energy release rate function.
        xc (List): Position of the crack tip.
    """
    # Store the model
    self.model = model
    # Store the crack tip position
    self.xc = xc
    # Set the critical energy release rate function
    self.Gc = jit(Gc_func)
    # Automatic differentiation of the objective function
    self.grad = jit(grad(self.objective))
    self.hess = jit(hessian(self.objective))
    # Automatic differentiation of the perturbed objective function
    self.grad_pert = jit(grad(self.objective_pert))
    self.hess_pert = jit(hessian(self.objective_pert))

objective(x, Ep, s, KIc, KIIc, Tc, KIp, KIIp, Tp, phi0)

Computes the objective function for the GMERR criterion.

This function computes the objective function for the GMERR criterion, which is used to find the optimal crack propagation angle and load factor.

Parameters:

Name Type Description Default
x ndarray

The optimization variables (crack propagation angle).

 required
Ep float

Plane strain modulus.

 required
s float

Internal length associated with T-stress.

 required
KIc float

Mode I stress intensity factor for the controlled problem.

 required
KIIc float

Mode II stress intensity factor for the controlled problem.

 required
Tc float

T-stress for the controlled problem.

 required
KIp float

Mode I stress intensity factor for the prescribed problem.

 required
KIIp float

Mode II stress intensity factor for the prescribed problem.

 required
Tp float

T-stress for the prescribed problem.

 required
phi0 float

Initial crack angle.

 required

Returns:

Name Type Description
float float

The value of the objective function.
Source code in src/gcrack/optimization_solvers.py 
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def objective(
    self,
    x: jnp.ndarray,
    Ep: float,
    s: float,
    KIc: float,
    KIIc: float,
    Tc: float,
    KIp: float,
    KIIp: float,
    Tp: float,
    phi0: float,
) -> float:
    """Computes the objective function for the GMERR criterion.

    This function computes the objective function for the GMERR criterion, which is used to find the optimal crack propagation angle and load factor.

    Args:
        x (jnp.ndarray): The optimization variables (crack propagation angle).
        Ep (float): Plane strain modulus.
        s (float): Internal length associated with T-stress.
        KIc (float): Mode I stress intensity factor for the controlled problem.
        KIIc (float): Mode II stress intensity factor for the controlled problem.
        Tc (float): T-stress for the controlled problem.
        KIp (float): Mode I stress intensity factor for the prescribed problem.
        KIIp (float): Mode II stress intensity factor for the prescribed problem.
        Tp (float): T-stress for the prescribed problem.
        phi0 (float): Initial crack angle.

    Returns:
        float: The value of the objective function.
    """
    # NOTE: The KIc (etc.) means controlled (not critical!)
    phi = x[0]
    # Compute the G star
    Gs_cc = G_star(phi, phi0, KIc, KIIc, Tc, Ep, s)
    Gs_cp = G_star_coupled(phi, phi0, KIc, KIIc, Tc, KIp, KIIp, Tp, Ep, s)
    Gs_pp = G_star(phi, phi0, KIp, KIIp, Tp, Ep, s)
    # Compute the Gc from phi
    gc = self.Gc(phi)
    # Compute and return the load factor
    delta = Gs_cp**2 - 4 * Gs_cc * (Gs_pp - gc)
    return (-Gs_cp + jnp.sqrt(delta)) / (2 * Gs_cc)

objective_pert(x, Ep, s, KIc, KIIc, Tc, KIp, KIIp, Tp, phi0)

Computes the perturbed objective function for the GMERR criterion.

This function adds a small perturbation to the objective function to avoid convergence to a maximum instead of a minimum.

Parameters:

Name Type Description Default
x ndarray

The optimization variables (crack propagation angle).

 required
Ep float

Plane strain modulus.

 required
s float

Internal length associated with T-stress.

 required
KIc float

Mode I stress intensity factor for the controlled problem.

 required
KIIc float

Mode II stress intensity factor for the controlled problem.

 required
Tc float

T-stress for the controlled problem.

 required
KIp float

Mode I stress intensity factor for the prescribed problem.

 required
KIIp float

Mode II stress intensity factor for the prescribed problem.

 required
Tp float

T-stress for the prescribed problem.

 required
phi0 float

Initial crack angle.

 required

Returns:

Name Type Description
float float

The value of the perturbed objective function.
Source code in src/gcrack/optimization_solvers.py 
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def objective_pert(
    self,
    x: jnp.ndarray,
    Ep: float,
    s: float,
    KIc: float,
    KIIc: float,
    Tc: float,
    KIp: float,
    KIIp: float,
    Tp: float,
    phi0: float,
) -> float:
    """Computes the perturbed objective function for the GMERR criterion.

    This function adds a small perturbation to the objective function to avoid
    convergence to a maximum instead of a minimum.

    Args:
        x (jnp.ndarray): The optimization variables (crack propagation angle).
        Ep (float): Plane strain modulus.
        s (float): Internal length associated with T-stress.
        KIc (float): Mode I stress intensity factor for the controlled problem.
        KIIc (float): Mode II stress intensity factor for the controlled problem.
        Tc (float): T-stress for the controlled problem.
        KIp (float): Mode I stress intensity factor for the prescribed problem.
        KIIp (float): Mode II stress intensity factor for the prescribed problem.
        Tp (float): T-stress for the prescribed problem.
        phi0 (float): Initial crack angle.

    Returns:
        float: The value of the perturbed objective function.
    """
    return (
        self.objective(x, Ep, s, KIc, KIIc, Tc, KIp, KIIp, Tp, phi0) + 1e-5 * x[0]
    )

solve(phi0, SIFs_controlled, SIFs_prescribed, s)

Solves for the optimal crack propagation angle and load factor.

This function uses gradient descent with line search to find the optimal crack propagation angle and load factor.

Parameters:

Name Type Description Default
phi0 float

Initial crack angle.

 required
SIFs_controlled dict

Stress intensity factors for the controlled problem.

 required
SIFs_prescribed dict

Stress intensity factors for the prescribed problem.

 required
s float

Internal length associated with T-stress.

 required

Returns:

Type Description

Tuple[float, float]: The optimal crack propagation angle and load factor.
Source code in src/gcrack/optimization_solvers.py 
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def solve(
    self, phi0: float, SIFs_controlled: dict, SIFs_prescribed: dict, s: float
):
    """Solves for the optimal crack propagation angle and load factor.

    This function uses gradient descent with line search to find the optimal
    crack propagation angle and load factor.

    Args:
        phi0 (float): Initial crack angle.
        SIFs_controlled (dict): Stress intensity factors for the controlled problem.
        SIFs_prescribed (dict): Stress intensity factors for the prescribed problem.
        s (float): Internal length associated with T-stress.

    Returns:
        Tuple[float, float]: The optimal crack propagation angle and load factor.
    """
    KIc, KIIc, Tc = (
        SIFs_controlled["KI"],
        SIFs_controlled["KII"],
        SIFs_controlled["T"],
    )
    KIp, KIIp, Tp = (
        SIFs_prescribed["KI"],
        SIFs_prescribed["KII"],
        SIFs_prescribed["T"],
    )

    # Perform the minimization
    kwargs = {
        "Ep": self.model.Ep_func(self.xc),
        "s": s,
        "KIc": KIc,
        "KIIc": KIIc,
        "Tc": Tc,
        "KIp": KIp,
        "KIIp": KIIp,
        "Tp": Tp,
        "phi0": phi0,
    }

    phi = gradient_descent_with_line_search(phi0, self.grad, kwargs=kwargs)

    # Check the stability of the solution (i.e., check if solution is a max)
    hess = self.hess([phi], **kwargs)[0][0]
    solution_is_max = hess < 0
    if solution_is_max:
        print("Found a maximum instead of minimum -> perturbating the objective")
        print("Note: this test might also be triggered by cups!")
        # Perform another gradient descent on the perturbed objective
        phi = gradient_descent_with_line_search(phi0, self.grad_pert, kwargs=kwargs)

    # Compute the load factor
    load_factor = self.objective([phi], **kwargs)

    return float(phi), float(load_factor)

Performs gradient descent with line search to minimize an objective function.

This function implements gradient descent with a custom line search to find the minimum of an objective function. It is designed to handle discontinuities and "cups" in the objective function.

Parameters:

Name Type Description Default
phi0 float

Initial guess for the crack propagation angle.

 required
gra Callable

The gradient of the objective function.

 required
tol float

Tolerance for convergence.

 1e-06
max_iter int

Maximum number of iterations.

 10000
kwargs dict

Additional arguments to pass to the gradient function.

 {}

Returns:

Name Type Description
float float

The optimal crack propagation angle.

Raises:

Type Description
RuntimeError

If the gradient descent fails to converge.
Source code in src/gcrack/optimization_solvers.py 
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def gradient_descent_with_line_search(
    phi0: float,
    gra: Callable,
    tol: float = 1e-6,
    max_iter: int = 10_000,
    kwargs: dict = {},
) -> float:
    """Performs gradient descent with line search to minimize an objective function.

    This function implements gradient descent with a custom line search to find the minimum
    of an objective function. It is designed to handle discontinuities and "cups" in the
    objective function.

    Args:
        phi0 (float): Initial guess for the crack propagation angle.
        gra (Callable): The gradient of the objective function.
        tol (float): Tolerance for convergence.
        max_iter (int): Maximum number of iterations.
        kwargs (dict): Additional arguments to pass to the gradient function.

    Returns:
        float: The optimal crack propagation angle.

    Raises:
        RuntimeError: If the gradient descent fails to converge.
    """
    print("│  │  Running the gradient descent with custom line search")
    # Initialization
    phi = float(phi0)
    converged = False
    for i in range(max_iter):
        # Determine the direction
        direction = -gra([phi], **kwargs)[0]
        # Check if the direction is close to 0
        if jnp.isclose(direction, 0):
            # Set a null increment
            dphi = 0
            # Set a null idx
            idx = 0
        else:
            # Apply line-search
            cs = [0.0] + [(jnp.pi / 2) ** k for k in range(-29, 2)]
            phis_test = jnp.array([phi + c * jnp.sign(direction) for c in cs])
            # Get the index associated with the first increase of the objective
            diff = jnp.array([gra([phi_test], **kwargs)[0] for phi_test in phis_test])
            # Create an array with the slope "in the direction of minimization"
            slope = jnp.sign(direction) * diff
            if jnp.all(slope < 0):
                # If the slope is always negative, take the largest step
                idx = -1
            elif jnp.all(slope > 0):
                # If the slope is always positive, take no step
                idx = 0
            else:
                # If the slope increases after a decrease, then local minimum
                idx = jnp.where(slope > 0)[0][0] - 1
                # If the first grad is positive, we are at the solution
                # This case only occurs when the grad is discontinuous (cups)
                if idx == -1:
                    idx = 0
            # Calculate the increment
            dphi = phis_test[idx] - phi
        # Update the solution
        phi += dphi
        # Generate an info message
        msg = "│  │  │  "
        msg += f"Step: {i + 1:06d} | "
        msg += f"phi: {jnp.rad2deg(phi):+7.2f}° | "
        msg += f"dphi: {abs(dphi):8.3g}"
        print(msg)
        # Check the convergence
        converged = idx == 0 or abs(dphi) <= tol
        if converged:
            print("│  │  │  Converged")
            break
        else:
            # Clip the angle phi
            phi = min(max(phi0 - 2 * jnp.pi / 3, phi), phi0 + 2 * jnp.pi / 3)

    # Check the convergence
    if not converged:
        raise RuntimeError(" └─ Gradient descent failed to converge!")
    return phi