Prediction of Fatigue Lifetime Using a Wavelet Transformation Induced Multi-time Scaling Method and Xfem

Abstract

Fatigue lifetime prediction due to dynamic crack growth is a significant issue for design and manufacture of engineering components. Damage accumulation in a material on the micro-scale is a main physical mechanism governing crack initiation. For high-cycle fatigue, the number of loading cycles leading to catastrophic fatigue failure can be of the order of millions. The simulation of such processes would be extremely expensive and time-consuming using conventional single time scale methods. In order to overcome this challenging requirement, a wavelet-transformation based multi-time scaling method is successfully adopted in this research to accelerate the prediction of accumulated damage for a large number loading cycles. In this work, the WATMUS technique is coupled with gradient- enhanced damage and the extended finite element method to simulate dynamic crack propagation for a turbine blade.

1 Introduction

To reduce the operating costs, many captital industrial components, such as turbine blades of aircraft engines, wind turbines or automotive components need regular maintenance and repair. In general, such engineering components are subjected to cyclic loading with high frequencies leading to severe fatigue failure. Fatigue crack initiation and propagation is a main physical mechanism governing the fatigue failure process. To predict the fatigue life of a component, fatigue tests are carried out by applying usually a constant amplitude cyclic load to measure the rate of crack growth. This process is expensive and time-consuming. Therefore in this present research, a numerical method for evaluation of the remaining lifetime of these individual components before and after the regeneration process is developed. This work is established in the collaborative research centre (CRC) 871 with the goal of predicting the functional properties of capital engineering components, like turbine blades. In the previous work, a new multiscale technique has been proposed for the efficient simulation of crack propagation and crack coalescence of macrocracks and microcracks (Holl et al. 2013). Contact formulation is implemented to enforce the non-penetration condition for crack surfaces (Kunin et al. 2017). In a gas turbine engine, air is compressed by the compressor. The temperature of air raises due to the compression and the combustion of fuel in the combustor. The air with high pressure and high temperature passes through the gas turbine. Then the turbine blades of air engines are responsible for transforming the energy of the pressurized exhaust gas to kinetic energy that can be used. The turbine blades are often the limiting components due to the complex and extreme loading conditions. An accurate prediction of the fatigue lifetime for a turbine blade is of high interest due to safety reasons and high replacement costs. The focus of this work lies on accurate and efficient simulation of fatigue failure due to dynamic crack growth under high-cycle loading combined with high thermal loads.

Nowadays, some numerical methods can be used to simulate crack growth. The finite element method (FEM) has been combined with linear elastic fracture mechanics (LEFM) for fatigue failure analysis (Colombo and Giglio 2006; Branco et al. 2015). The boundary element method (BEM) is also an alternative tool for crack simulations (Peng et al. 2017; Portela et al. 1992; Yan 2006). In recent years, phase- field models have gained more attention in fracture modeling. A severe drawback of FEM, as well as the phase-field models for crack simulation is the necessity for severe mesh refinement near the crack surfaces. For FEM, a remeshing process around the crack front is necessary as the crack grows, leading to poor computational efficiency. Hence, dealing with the possible inaccuracy and high computational costs still remains a challenging requirement for crack simulation using FEM. In this work, the extended finite element method (XFEM) (Belytschko and Black 1999; Moës et al. 1999), which was developed by adding proper enrichment functions to the finite element approximations using the partition of unity framework, is adopted. Level set techniques are frequently used to handle the geometric update of crack configurations during crack propagation (Gravouil et al. 2002; Oliver et al. 2004; Sukumar et al. 2003).

The formation of cracks on the macro-scale is a result of damage accumulation in a material on the micro-scale. In continuum damage mechanics, a scalar damage variable is often introduced to describe the loss of material integrity and the reduction of material stiffness. An assumption is that a crack starts to initiate when the accumulated damage in a material reaches a critical value. The results of early finite element computations including a local damage model show severe mesh sensitivity. This can be explained as the damage in material always localizes in the weaker part of the material. To overcome these shortcomings, a gradient-enhanced damage formulation (Peerlings et al. 2000) has been developed for finite element solutions of quasi-brittle and fatigue fractures, and therefore will be adopted in this research. A gradient-enhanced damage model combined with the XFEM is developed to simulate crack growth with the consideration of thermal loading. Dynamics is taken into account by considering inertial effects. The key issue in the prediction of the high-cycle fatigue life is the efficient computation of a large number of loading cycles in a reasonable time. With a conventional single time scale method, each loading cycle is divided into a number of small time steps to ensure an appropriate accuracy. The huge number of time steps in the fatigue life prediction can make the problems nearly unsolvable within a reasonable amount of time. A wavelet transformation induced multi-time scaling (WATMUS) algorithm is applied to improve the computational efficiency for prediction of damage evolution after millions of loading cycles (Joseph et al. 2010; Chakraborty and Ghosh 2013; Yaghmaie et al. 2016; Ghosh and Chakraborty 2013). This method benefits from the split of time into two scales: (i) a fine time scale τ with a rapidly oscillating high-frequency response within a complete cycle, and (ii) a coarse time scale with a low-frequency material response over the entire loading process. The material response in the coarse time scale is a function of the cycle number N, and hence behaves monotonically. The information of fine-scale response of material variables is transformed to the computation and modification of the wavelet coefficients by means of discrete wavelet transformation, and the cycle scale material response is evaluated by appropriate coarse scale rate equations. The coarse scale evolution equation is connected to the fine scale solutions. The contribution presented in this article is the combination of WATMUS, gradient-enhanced damage and XFEM. The evolution of the local damage in the coarse time scale is accurately predicted using the “cycle jump” algorithm in WATMUS. A significant speed-up is obtained comparing the single time scale method.

This paper is organized as follows: In Sect. 2, the elasticity-based damage evolution law for high-cycle fatigue is introduced along with the governing equations and the gradient enhancement of the damage model. A simulation of thermo-mechanical dynamic crack propagation for a turbine blade is demonstrated. In Sect. 3, the mathematical details of the WATMUS method are given including a numerical example to show the high efficiency and accuracy of the WATMUS method for prediction of high-cycle fatigue damage evolution. Finally in Sect. 4 we make conclusions of the research in our subproject within the program of CRC 871, and discuss the possibility of future application of this work.

2 Thermo-mechanical Dynamic Fatigue Crack Growth Using XFEM

Duflot (2008) applied the XFEM for the simulation of steady-state thermo-elastic fractures. This idea can be extended to three-dimensional dynamic crack propagation analysis with transient heat conduction.

2.1 Governing Equations

Due to the high frequency of the external load, the plastic strain can be neglected compared to the elastic part. Hence, a linear-elastic, isotropic material is assumed in this work. Consider a thermally and mechanically loaded, cracked solid in a domain Ω bounded by Γ, the governing equations for damage-based thermo-mechanical dynamic problems with the assumption of small deformation can be defined by

$$\rho {c}_{p}\dot{T} + \nabla \cdot {\bf q} = 0, {\bf q} = -k\nabla T \,\, \text{in} \,\, \Omega$$

(1)

$$\nabla \cdot \boldsymbol{\sigma} +{\bf f}=\rho \ddot{{\bf u}}\text{ in }\Omega$$

(2)

$$\Phi \; - \;c\Delta \Phi \; = \;\tilde{D}\;\text{in} \;\Omega$$

(3)

$$\boldsymbol{\sigma} =\left(1-\Phi \right){\mathbb{D}} :\left(\boldsymbol{\varepsilon} -{\boldsymbol{\varepsilon} }_{\text{T}}\right),\boldsymbol{\upvarepsilon} ={\nabla }_{\text{s}}{\bf u}, {\boldsymbol{\varepsilon} }_{T}=\alpha \left(T-{T}_{0}\right){\bf I}$$

(4)

In these equations, u is the displacement field, $\ddot{{\bf u}}$ is the acceleration, T is the temperature field, heat flux vector is given as q, stress tensor is σ, strain tensor is ε, thermal strain tensor is defined by ε_T with the initial temperature field T₀, local damage is given as $\tilde{D}$ , and gradient-enhanced damage field ϕ. There are a variety of damage models to compute the local damage $\tilde{D}$ for brittle and ductile materials. The details about the damage model used in this reseach will be introduced in the next section. The gradient-enhanced damage Eq. 3 proposed in Boggess and Narcowich (2011) is strongly coupled with the displacement and the temperature field through Eq. 4. The material properties include the elastic fourth-order Hooke’s tensor $\mathbb{D}$, thermal conductivity k, expansion coefficient α, and c is a nonlocality parameter that can be defined by c = l²/2, where l denotes the internal length of the nonlocality. We assume that there is no body force f = 0. I is second-order identity tensor and ∇_s denotes the symmetric gradient operator.

2.2 Elasticity Based High-Cycle Fatigue Damage

The cyclic external loading causes a repetitive accumulation of fatigue damage, which leads to initiation, growth and coalescence of cracks at the mesoscale and macroscale levels. The existing experiments of high-cycle fatigue revealed that the damage increases slowly at the beginning of the fatigue process, and then growth of damage accelerates until a sudden loss of stiffness can be observed near the end of the fatigue analysis. To capture this evolving property, the rate equation of high-cycle fatigue damage growth suggested by Peerlings et al. (2000) is used, which can be written as

$$\dot{\tilde{D }}= \left\{\begin{array}{c}g\left(\tilde{D },\stackrel{\sim }{\varepsilon }\right)\dot{\stackrel{\sim }{\varepsilon }}, \text{if}\, f \ge 0,\dot{f} \ge 0 \,\, \text{and} \,\, \tilde{D }<1\\ 0, \text{else} \end{array}\right.$$

(5)

with the evolution function expressed by

$$g\left(\tilde{D },\stackrel{\sim }{\varepsilon }\right)=C{e}^{\alpha \tilde{D } }{\stackrel{\sim }{\varepsilon }}^{\beta }$$

(6)

In this equation, C, α and β are material constants, f is a loading function that can be defined by

$$f\left(\stackrel{\sim }{\varepsilon }, \kappa \right) \, = \, \stackrel{\sim }{\varepsilon }-{\kappa }_{0}$$

(7)

where $\tilde{\varepsilon}$ is the equivalent strain defined in terms of the strain tensor, and κ₀ is given as the threshold value defining the elastic domain. It can be seen that if the equivalent strain during the loading remains smaller than κ₀, the material is always located in the elastic region and no damage development can be observed in all loading cycles. In this case, the fatigue failure will never occur. The condition f ≥ 0 implies that the damage can increase only when the equivalent strain is outside of the elastic domain. Additionally, the damage variable is allowed to develop only during a loading process, i.e. $\dot{f}\; \ge \,0$.

2.3 Discretization with the Extended Finite Element Method

Based on the discretization scheme in FEM, the displacement field in the XFEM (Belytschko and Black 1999; Gravouil et al. 2002) is described by introducing additional degrees of freedom to the nodes and enrichment functions to capture the discontinuity and singularity of the elements intersected by the cracks. The discretized form of the displacement field in the XFEM can be given as

$${{\bf u}}^{h}\left({\bf x}\right)= \sum_{I \epsilon \mathcal{I} }{N}_{I}\left({\bf x}\right){{\bf u}}_{I}+ \sum_{J \epsilon \mathcal{J}}{N}_{J}\left({\bf x}\right)H\left({\bf x}\right){{\bf a}}_{J}+\sum_{K \epsilon \mathcal{K}}\sum_{j=1}^{4}{N}_{K}\left({\bf x}\right){f}_{j}\left({\bf x}\right){{\bf b}}_{Kj}$$

(8)

The first term is the same as the standard FE displacement approximation and therefore the nodal subset I contains all nodes. The second and third terms are the additional parts enriched by the Heaviside function H(x) and some crack tip enrichment functions f_j(x). Generally, four crack tip enrichment function stemming from linear elastic fracture mechanics can be used:

$${f}_{1-4 }\left({\bf x}\right)= \left\{\sqrt{r}\text{sin}\left(\frac{\uptheta }{2}\right), \sqrt{r}\text{cos}\left(\frac{\uptheta }{2}\right), \sqrt{r}\text{sin}\left(\frac{\uptheta }{2}\right)sin\theta , \sqrt{r}\text{sin}\left(\frac{\uptheta }{2}\right)cos\theta \right\}$$

(9)

where (r, θ) represent the polar coordinate system with a center at the crack tip. The nodal subset $J\; \subset \;\mathcal{I}$ in Eq. 8 contains all nodes of the elements that are completely cut by the crack, and the nodal subset $\mathcal{K}\; \subset \;\mathcal{I}$ contains all nodes of the elements in which the crack tip or the crack front is located. The Heaviside function is defined by

$$H(\textbf{x})\; = \;\left\{ {\begin{array}{*{20}c} 1: & {\varphi (\textbf{x}) \ge 0} \\ { - 1:} & {\varphi (\textbf{x}) < 0} \\ \end{array} } \right.$$

(10)

where $\varphi (\textbf{x})$ is the signed distance function from x to the crack surface. Due to the geometric discontinuity caused by the cracks, the temperature field and the gradient- enhanced damage are also discontinuously distributed across the cracks. Singularity of the temperature gradient at the crack tip or crack front can appear as well. To fulfill these requirements, both the temperature field and the gradient enhanced damage are approximated similar to the displacement field by

$${T}^{h}\left(\textbf{x}\right)= \sum_{I \epsilon \mathcal{I}}{\overline{N} }_{I}\left(\textbf{x}\right){T}_{I}+\sum_{J \epsilon \mathcal{J}}{\overline{N} }_{J}\left(\textbf{x}\right)H\left(\textbf{x}\right){a}_{J}^{T}+ \sum_{K \epsilon \mathcal{K}}\sum_{j=1}^{4}{\overline{N} }_{K}\left({\bf x}\right){f}_{j}\left(\textbf{x}\right){b}_{Kj}^{T}$$

(11)

$${\Phi }^{h}\left(\textbf{x}\right)= \sum_{I \epsilon \mathcal{I}}{\tilde{N}}_{I}\left(\textbf{x}\right){\Phi }_{I}+\sum_{J \epsilon \mathcal{J}}{\tilde{N }}_{J}\left(\textbf{x}\right)H\left(\textbf{x}\right){a}_{J}^{\Phi }+ \sum_{K \epsilon \mathcal{K}}\sum_{j=1}^{4}{\tilde{N }}_{K}\left(\textbf{x}\right){f}_{j}\left(\textbf{x}\right){b}_{Kj}^{\Phi }$$

(12)

If only the first crack tip enrichment function $f_{1} ({\bf x})\; = \;\sqrt r \;\sin (\theta /2)$ is used to reduce the number of unknowns for the coupled problem and significantly decrease the condition number of the resulting coefficient matrices (Legrain et al. 2005; Loehnert et al. 2011), these approximations can be written in a more compact matrix form

$${\textbf{u}}^{h}\left(\textbf{x}\right)= \sum_{I}[\begin{array}{ccc}{N}_{I}& {N}_{I}H& {N}_{I}{f}_{1}\end{array}]\left[\begin{array}{c}{\textbf{u}}_{I}\\ {\textbf{a}}_{I}\\ {\textbf{b}}_{I1}\end{array}\right]= \sum_{I}{\textbf{N}}_{I}{\overline{\textbf{u}} }_{I}$$

$${T}^{h}\left(\textbf{x}\right)=\sum_{I}{\overline{\textbf{N}} }_{I}{\overline{\textbf{T}}}_{I}, {\Phi}^{h}\left(\textbf{x}\right)=\sum_{I}{\tilde{\textbf{N}}}_{I}{\overline{\boldsymbol{\Phi}}}_{I}$$

(13)

The displacement field, temperature and gradient enhanced damage are strongly coupled with each other. We consider the weak form of the governing equations for all fields supplemented by the associated boundary conditions. The governing Eqs. (1–3) are multiplied by their corresponding test functions δT , δu and δϕ, respectively, and subsequently integrated in the domain Ω. Using the divergence theorem, the resulting equation leads to the weak form:

$$\textbf{R}_{\textbf{u}} \; = \;\int_{\Omega } {\delta \mathbf{\varepsilon} \;:\;\mathbf{\sigma} d\Omega } \; + \;\int_{\Omega } {\delta \textbf{u}\; \cdot \;\rho \ddot{\textbf{u}}d\Omega } \; - \;\int_{\Gamma } {\delta \textbf{u}\; \cdot \;\tilde{\textbf{t}}d\Gamma } \;\mathop = \limits^{!} \;0$$

$${{\bf R}}_{\Phi }={\int}_{\Omega }\delta\Phi \left(\tilde{D }-\Phi \right)\text{d}\Omega-\text{c}{\int}_{\Omega }\boldsymbol{\nabla} \delta\Phi \cdot \boldsymbol{\nabla} \Phi \text{d}\Omega \,\, \mathop = \limits^{!} \,\, 0$$

$${\bf R}_{T} = \mathop \int \limits_{{\Omega }} \delta T \rho c_{p} \dot{T}{\text{d}}\Omega - \mathop \int \limits_{{\Omega }} \left( {\boldsymbol{\nabla} \delta T} \right) \; \cdot \;{{\bf q}\text{d}}\Omega + { }\mathop \int \limits_{{{\Gamma }_{{\text{q}}} }} \delta T{\bf q} \cdot {\bf n}{\text{d}}\Gamma \,\, \mathop = \limits^{!} \,\, 0 \,\, { }$$

(14)

Applying the Bubnov-Galerkin method and after the complete discretization of u, $\ddot{{\bf u}}$ , T and ϕ with XFEM ansatz functions, the set of non-linearly coupled equations in each iteration step of the Newton-Raphson procedure can be obtained

$$\begin{aligned} & \left[ {\begin{array}{*{20}c} {{\bf M}_{uu} } & {\bf 0} & {\bf 0} \\ {\bf 0} & {\bf 0} & {\bf 0} \\ {\bf 0} & {\bf 0} & {\bf 0} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\Delta \ddot{\overline{{\bf u}}}} \\ {\Delta \ddot{\overline{\boldsymbol{\Phi} }}} \\ {\Delta \ddot{\overline{{\bf T}}}} \\ \end{array} } \right]\; + \left[ {\begin{array}{*{20}c} {\bf 0} & {\bf 0} & {\bf 0} \\ {\bf 0} & {\bf 0} & {\bf 0} \\ {\bf 0} & {\bf 0} & {{\bf C}_{TT} } \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\Delta \dot{\overline{{\bf u}}}} \\ {\Delta \dot{\overline{\boldsymbol{\Phi} }}} \\ {\Delta \dot{\overline{{\bf T}}}} \\ \end{array} } \right]\;\; \\ & + \;\left[ {\begin{array}{*{20}c} {{\bf K}_{uu} } & {{\bf K}_{u\Phi } } & {{\bf K}_{uT} } \\ {{\bf K}_{\Phi u} } & {{\bf K}_{\Phi \Phi} } & {\bf 0} \\ {\bf 0} & {\bf 0} & {{\bf K}_{TT} } \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\Delta \overline{{\bf u}}} \\ {\Delta \overline{\boldsymbol{\Phi} }} \\ {\Delta \overline{{\bf T}}} \\ \end{array} } \right]\; = \;\left[ {\begin{array}{*{20}c} {{\bf R}_{{\bf u}} } \\ {{\bf R}_{\Phi } } \\ {{\bf R}_{T} } \\ \end{array} } \right]. \\ \end{aligned}$$

(15)

with the definitions of the matrices

$${{\bf M}}_{uu}= {\int}_{\Omega }{{\bf N}}^{T}\rho {\bf N}\text{d} \Omega$$

$${{\bf C}}_{\theta \theta }= {\int}_{\Omega }{\bf N}\rho {c}_{p}{{\bf N}}^{T}\text{d}\Omega$$

$${{\bf K}}_{uu}= {\int}_{\Omega }\left(1-\Phi \right){{\bf B}}^{T}{\bf D}{\bf B}\text{d}\Omega$$

$${{\bf K}}_{u\Phi }= - {\int}_{\Omega }{{\bf B}}^{T}{\bf D}\left(\boldsymbol{\varepsilon} -{\boldsymbol{\varepsilon} }_{T}\right){\tilde{{\bf N} }}^{T}\text{d}\Omega$$

$${{\bf K}}_{u\text{T}}= - {\int}_{\Omega }\left(1-\Phi \right){{\bf B}}^{T}{\bf D}\alpha \tilde{{\bf I} }{\overline{{\bf N}} }^{T}\text{d}\Omega$$

$${{\bf K}}_{\Phi\text{u}}= - {\int}_{\Omega }\tilde{{\bf N} }q{{\bf s}}^{T}{\bf B}\text{d}\Omega$$

$${{\bf K}}_{{\Phi \Phi }}= {\int}_{\Omega }\left[\tilde{{\bf N} }{\tilde{{\bf N} }}^{T}+{\tilde{{\bf B} }c\tilde{{\bf B} }}^{T}\right]\text{d}\Omega$$

$${{\bf K}}_{TT}= {\int}_{\Omega }\overline{{\bf B} }k{\overline{{\bf B}} }^{T}\text{d}\Omega$$

(16)

where q is given by Peerlings et al. (2000)

$$q= \frac{\left(1-\theta \right)g\left({D}_{n}, {\varepsilon }_{n}\right)+ \theta g\left({D}_{n+1}^{\left(i\right)},{\varepsilon }_{n+1}^{\left(i\right)}\right)+\theta \frac{\partial g}{ \partial \varepsilon }{\left.\right|}_{n+1}^{\left(i\right)}\left({\stackrel{\sim }{\varepsilon }}_{n+1}^{\left(i\right)}-{\stackrel{\sim }{\varepsilon }}_{n}\right)}{1-\theta \frac{\partial g}{ \partial D }{\left.\right|}_{n+1}^{\left(i\right)}\left({\stackrel{\sim }{\varepsilon }}_{n+1}^{\left(i\right)}-{\stackrel{\sim }{\varepsilon }}_{n}\right)}$$

(17)

and the column vector s is a partial derivate ${\bf s}\; = \;\partial \tilde{\varepsilon }\;/\;\partial \boldsymbol{\varepsilon}$. These equations can be solved using the Newmark-beta method.

2.4 Dynamic Crack Propagation Using the Level Set Methods

Employing the XFEM for crack propagation simulations, the evolving geometry of the crack needs to be tracked. A crack front in the three-dimensional space can be determined by interaction of two orthogonal signed distance functions, also called level sets, that is, the crack surface level set φ and the crack front level set Φ. In this work, the level set method (Gravouil et al. 2002) and a fast marching method (FMM) (Sukumar et al. 2003) are combined for tracking the growth of three-dimensional cracks implicitly. The velocity for crack growth is extended by solving two Hamilton Jacobi equations to steady state in the level set subdomain simultaneously.

2.4.1 Update of ϕ Using a Global Crack Tracking Method

Let v represent the velocity of crack surface extension, ${\bf n}_{\phi } \; = \;\nabla{\phi }$ is a normal vector perpendicular to the iso-zero surface of the level set $\phi$ . Two independent vectors T and S on the updated crack surface can be defined by ${\bf T}\; = \;{\bf v} \;\;\text{and}\;\;{\bf S}\; = \;\frac{\nabla \psi \; \times \;\nabla \phi }{{\left\| {\nabla \psi \; \times \;\nabla \phi } \right\|}}$. It can be seen that the vectors T and S are perpendicular to the normal vector ${\bf n}_{\phi }$ of the searched surface. The geometry update of the crack surface is equivalent to the solution of ${\bf T}\; \cdot \;\nabla \phi \; = \;{\bf S} \cdot \nabla \phi \; = \;0$. Here we employ the global crack tracking method (Oliver et al. 2004) to solve the extension of velocity more efficiently. Multiplying the equations ${\bf T}\; \cdot \;\nabla \phi \; = \;0\;\;\mbox{and}\;\;{\bf S} \cdot \nabla \phi \; = \;0$ by T and S, respectively, we obtain:

$${\bf T}\otimes {\bf T}\cdot \nabla \phi ={\bf S}\otimes {\bf S}\cdot \nabla \phi ={\bf 0}$$

(18)

The solution of the following boundary value problem in the level set subdomain satisfies the differential Eq. 18:

$$\nabla \cdot {\bf q}=0, {\bf q}=-\mathbf{\mathcal{K}}\cdot \nabla \phi \,\, \text{in} \,\, \Omega$$

(19)

$$\overline{q} \equiv {\bf q} \cdot {\bf n} \,\, \text{on } \,\, \partial\Omega$$

(20)

$$\mathbf{\mathcal{K}}= {\bf T}\otimes {\bf T}+{\bf S}\otimes {\bf S}+\varepsilon {\bf I}$$

(21)

where εI is an artificial treatment to avoid the singularity of the tensor K.

2.4.2 Update of Ψ by Solving Hamilton–Jacobi Equation

To track the motion of crack front, the scheme known as the Hamilton–Jacobi equation of motion can be utilized

$$\frac{\partial \psi }{\partial t}+{\bf v} \cdot \nabla \psi =0$$

(22)

where v is the velocity of crack propagation.

2.4.3 Reinitialization of Ψ and ϕ Using FMM

A signed distance function must satisfy the Eikonal equation

$$\Vert \nabla \phi \Vert =1$$

(23)

When the Hamilton-Jacobi equations are solved numerically, the reinitialization of ψ and $\phi$ by the FMM (Sukumar et al. 2003) can ensure the Eikonal equation to be almost fulfilled, which can prevent numerical instabilities.

2.4.4 Reorthogonalization of Ψ and ϕ with Embedded Reinitialization

After updating the level set function ψ, it needs to be orthogonalized to $\phi$ by solving the following Hamilton–Jacobi equation

$${R}_{e}=\frac{\partial \psi }{\partial t }+sign\left(\phi \right)\frac{\nabla \phi }{\Vert \nabla \phi \Vert } \cdot \nabla \psi =0$$

(24)

It can be seen that, the orthogonality condition ∇$\phi$ · ∇ψ = 0 is fulfilled automatically when the above equation is solved to the steady state. The orthogonalization process by solving the Eq. 24 may destroy the signed distance property of ψ. To ensure the signed distance property, the Eikonal equation ||∇ψ|| = 1 is embedded into Eq. 24 by means of a weak form. The Hamilton–Jacobi equation can be solved using the Galerkin Least Square (GLS) method (Beese et al. 2018). The potential of the least square part is defined by

$${\Pi }^{LS}=\frac{1}{2}{\int}_{\Omega }\tau {R}_{e}^{2}\text{d}\Omega$$

(25)

The stabilization parameter τ is dependent on the element size. For the determination and physical interpretation of τ the reader is referred to Barth and Sethian (1998) and Beese et al. (2018). The Eikonal equation is enforced by adding a penalty term. The potential of the penalty part is given by

$${\Pi }^{\text{PE}}=\frac{1}{2}{\int}_{\Omega }\gamma {(\Vert \nabla \psi \Vert -1)}^{2}\text{d}\Omega$$

(26)

where γ is the penalty parameter. Then the weak form can be obtained by

$$\delta\Pi =\delta {\Pi }^{G}+\delta {\Pi }^{LS}+\delta {\Pi }^{PE}=0$$

(27)

where

$$\delta {\Pi }^{G}= {\int}_{\Omega }\delta \psi {R}_{e}\text{d}\Omega ={\int}_{\Omega }\delta \psi \left(\frac{\partial\uppsi }{\partial\uptau }+sign\left(\psi \right)\frac{\nabla \phi }{\Vert \nabla \phi \Vert }\cdot \nabla \psi \right) \text{d}\Omega$$

$$\begin{aligned} & \delta {\Pi }^{LS}={\int}_{\Omega }\delta {R}_{e}\uptau {\text{R}}_{\text{e}}\text{d}\Omega \\ & ={\int}_{\Omega }\left(\frac{\partial \delta \psi }{\partial\uptau }+\text{sign}\left(\upphi \right)\frac{\nabla \phi }{\Vert \nabla \phi \Vert }\cdot \nabla\updelta \psi \right)\tau \left(\frac{\partial \psi }{\partial\uptau }+\text{sign}\left(\upphi \right)\frac{\nabla \phi }{\Vert \nabla \phi \Vert }\cdot \nabla \psi \right)\text{d}\Omega \\ \end{aligned}$$

$$\delta {\Pi }^{PE}={\int}_{\Omega }\delta (\Vert \nabla \psi \Vert )\gamma \left(\Vert \nabla \psi \Vert -1\right)\text{d}\Omega ={\int}_{\Omega }\gamma \left(1-\frac{1}{\Vert \nabla \psi \Vert }\right)\nabla \psi \cdot (\nabla \delta \psi )\text{d}\Omega$$

(28)

The above formulation can be discretized with the standart finite element procedure.

2.5 Numerical Results of Dynamic Crack Propagation

To demonstrate the applicability of the presented model to dynamic crack propagation problems, a turbine blade with a predefined crack at the front edge, as shown in Fig. 1, is simulated. A turbine blade is subjected to high centrifugal force with thermal loading up to 1400 ℃. The dynamic propagation of a predefined crack under high-frequency centrifugal force is investigated with the developed model. Due to the complexity of the geometry, the turbine blade is discretized with ten-node tetrahedral elements with quadratic shape functions. The simulated distribution of the gradient enhanced damage field after a few steps of crack propagation is shown in Fig. 2.

Fig. 1

Mesh of a turbine blade with a predefined edge crack

Fig. 2

Simulation results of the gradient enhanced damage field with dynamic crack propagation

3 Review of the Wavelet Transformation Induced Multi-time Scaling Method (WATMUS)

The wavelet transformation based multi-time scaling (WATMUS) methodology has been successfully developed and implemented to accelerate the time integration in crystal plasticity finite element analysis for a large number of loading cycles Joseph et al. (2010); Chakraborty and Ghosh (2013); Yaghmaie et al. (2016); Ghosh and Chakraborty (2013). In this methodology, the continuous time is split into two scales: a coarse time scale identified with the cycle number N, and a fine time scale τ within each cycle. The value of any state variable α₀ at the beginning of a given cycle N can be thought of as a coarse time scale variable. This state variable will not vary in the fine τ-scale within each cycle and hence, it can be considered as a function of the cycle number N, i.e. α₀ = α₀(N). The objective is to find a coarse time scale (cycle scale) evolution equation in the form

$$\frac{\text{d}{\alpha }_{0}(N)}{\text{d}N}=f\left({\alpha }_{0}\left(N\right),{u}_{i,k }^{\alpha }\left(N\right)\right)$$

(29)

where u^α are the wavelet coefficients for the i-th displacement component of node α. Thus, a two-scale representation of any given variable β can be obtained in terms of the orthogonal wavelet basis functions as

$$\beta \left(t\right)=\beta \left(N,\tau \right)=\sum_{k=1}^{n}{\beta }^{k}\left(N\right){\psi }_{k}\left(\tau \right)$$

(30)

The coefficients β^k depend only on the cycle number N. The wavelets are defined in a finite domain and have compact support. Hence, the sudden change of a material response can be well captured by wavelet-based solutions without the Gibbs phenomena that occurs in the Fourier transformation based solutions. The multiresolution analysis represents a square integrable function in continuous time at different resolutions, and hence an optimal number of coefficients needs to be solved.

3.1 Fundamentals of Wavelets

The wavelet multiresolution analysis (Strang and Nguyen 1997; Boggess and Narcowich 2011) is based on a nested sequence of subspaces at different resolutions

$$\cdots \subset {V}_{-1}\subset {V}_{0}\subset {V}_{1}\subset {V}_{2}\subset \dots \subset {V}_{m}\subset \dots \subset {L}^{2}({\mathbb{R}})$$

(31)

Each subspace V_m is spanned by a set of basis functions constructed from dilations and translations of the scaling function $\phi$ (τ) defined by:

$${\phi }_{m,n}\left(\tau \right)={2}^\frac{m}{2}\phi ({2}^{m}\tau -n)$$

(32)

where m denotes the number of resolution and n specifies the number of transition. The multiresolution analysis needs an orthogonal complementary space denoted by W_m. The subspace W_m is spanned by a set of replicas constructed from a single mother wavelet ψ(τ) at the resolution level m, that is:

$${\psi }_{m,n }\left(\tau \right)= {2}^\frac{m}{2}\psi ({2}^{m}\tau -n)$$

(33)

Any square integrable function f (τ) can be approximated by its projection onto each subspace V_m and W_m as

$${f}_{m}\left(\tau \right)= \sum_{n}{c}_{m,n}{\phi }_{m,n}\left(\tau \right), \textit{w}_{m}\left(\tau \right)=\sum_{n}{d}_{m,n}{\psi }_{m,n}\left(\tau \right)$$

(34)

Since V₀ ⊂ V₁, the scaling function $\phi$ (τ) ∈ V₀ can be expanded in terms of the basis that spans V₁ as

$$\phi \left(\tau \right)= \sum_{k}{h}_{k}{2}^\frac{1}{2}\phi (2\tau -k)$$

(35)

where h_k corresponds to the low pass filter coefficients. Eq. 35 is known as the dilation or refinement equation. Moreover, W₀ is spanned by the wavelet function ψ(τ) and its translates ψ(τ − n). Since ψ(τ) ∈ W₀ ⊂ V₁, in analogy to Eq. 35, we have an expansion of ψ(τ) as

$$\psi \left(\tau \right)= \sum_{k}{g}_{k}{2}^\frac{1}{2}\phi (2\tau -k)$$

(36)

where g_k represents the high pass filter coefficients. Filter coefficients for the Daubechies-4 wavelet used in this work (Strang and Nguyen 1997; Boggess and Narcowich 2011) are given as

$${h}_{0}=\frac{1+\sqrt{3}}{4\sqrt{2}}, {h}_{1}=\frac{3+\sqrt{3}}{4\sqrt{2}}, {h}_{2}=\frac{3-\sqrt{3}}{4\sqrt{2}}, {h}_{3}=\frac{1-\sqrt{3}}{4\sqrt{2}}$$

(37)

$${g}_{0}=\frac{1-\sqrt{3}}{4\sqrt{2}}, {g}_{1}=-\frac{3-\sqrt{3}}{4\sqrt{2}}, {g}_{2}=\frac{3+\sqrt{3}}{4\sqrt{2}}, {g}_{3}=-\frac{1+\sqrt{3}}{4\sqrt{2}}$$

(38)

Based on the decomposition V_m+1 = V_m ⊕ W_m, the approximation of an arbitrary function f (τ) ∈ L²(R) at the resolution of m + 1, that is, f_m+1(τ), can be obtained by adding the lower-level approximation f_m(τ) and its orthogonal complement w_m(τ).

$$f_{{m + 1}} (\tau )\; = \;\sum\limits_{l} {c_{{m + 1,l}} \phi _{{m + 1,l}} (\tau )\; = \;} \underbrace {{\sum\limits_{n} {c_{{m,n}} \phi _{{m,n}} (\tau )} }}_{{f_{m} (\tau )}}\; + \;\underbrace {{\sum\limits_{n} {d_{{m,n}} \psi _{{m,n}} (\tau )} }}_{{w_{m} (\tau )}}\;$$

(39)

Using the above equations and defining $a_{m,n} = 2^{\frac{m}{2}} c_{m,n}$ and $b_{m,n} = 2^{\frac{m}{2}} d_{m,n}$, the analysis equation can be obtained

$$\left[\begin{array}{c}{{\bf a}}_{m-1}\\ {{\bf b}}_{m-1}\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{{\bf H}}_{m}\\ {{\bf G}}_{m}\end{array}\right]{{\bf a}}_{m}$$

(40)

where

$$\left[\begin{array}{c}{{\bf H}}_{m}\\ {{\bf G}}_{m}\end{array}\right]=\left[\begin{array}{ccccccccc}{h}_{0}& {h}_{1}& {h}_{2}& {h}_{3}& 0& 0& \dots & 0& 0\\ 0& 0& {h}_{0}& {h}_{1}& {h}_{2}& {h}_{3}& \dots & 0& 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {h}_{2}& {h}_{3}& 0& 0& 0& 0& \dots & {h}_{1}& {h}_{2}\\ {g}_{0}& {g}_{1}& {g}_{2}& {g}_{3}& 0& 0& \dots & 0& 0\\ 0& 0& {g}_{0}& {g}_{1}& {g}_{2}& {g}_{3}& \dots & 0& 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {g}_{2}& {g}_{3}& 0& 0& 0& 0& \dots & {g}_{1}& {g}_{2}\end{array}\right]$$

(41)

This is the essential tool for obtaining the discrete wavelet transformation (DWT). The sampled values of a function a_m = f = [f₀ f₁ f₂ … f_Np-1], where N_p = 2 ^m, are transformed to all levels of wavelet coefficients and the approximation coefficients of the lowest resolution through the DWT

$${\bf c}={\bf T} {\bf f}$$

(42)

The vector f with dimension N = 2^m is defined by sampling the function f (t) at N equally spaced points. T is an N × N orthogonal transformation matrix. Once a DWT has been performed on the given sampled signal, the detail coefficients can be analyzed and modified. The modified signal can be reconstructed through an inverse DWT by

$${{\bf f}}_{mod}={{\bf T}}^{T}{{\bf c}}_{mod}$$

(43)

3.2 Solving Coarse Time-Scale Evolution Equations

An implicit two-step second order backward difference scheme (BDF2) has been proposed to integrate the coarse scale variable over a number of cycles (Joseph et al. 2010), for which the residual at the N-th cycle is

$$\begin{aligned} r_{c} \left( {{{\bf x}},N} \right)\; & = \;\left( {1\; + \;\beta _{3} {{\Delta }}N} \right)\alpha _{0} \left( {{{\bf x}},\;N} \right) - \beta _{1} \alpha _{0} \left( {{{\bf x}},N - {{\Delta }}N} \right) \\ & - \beta _{2} \alpha _{0} \left( {{{\bf x}},~N\; - \;{{\Delta }}N\; - \;{{\Delta }}N_{p} } \right)\; - \;\beta _{3} {{\Delta }}N\alpha _{0} ({\bf x},\;N\; + \;1)\;\mathop = \limits^{!} \;~0 \\ \end{aligned}$$

(44)

where the parameters are

$${\beta }_{1}=\frac{{\left(1+r\right)}^{2}}{{\left(1+r\right)}^{2}-1}, {\beta }_{2}=-\frac{1}{{\left(1+r\right)}^{2}-1}, {\beta }_{3}= \frac{{\left(1+r\right)}^{2}-r-1}{{\left(1+r\right)}^{2}-1} \,\, \text{and} \,\, r=\frac{\Delta {N}_{p}}{\Delta N}$$

(45)

and α_Np (x, N) is the internal variable at the last sampling point of cycle N. ∆N and ∆N_p are cycle jumps of the current and previous steps. Eq. 44 is solved using the Newton-Raphson iterative method for the value of α₀(N). The Jacobian corresponding to the above residual Eq. 44 can be obtained as

$$\frac{{\partial r}}{{\partial \alpha _{0} }}({\bf x},\,N)\; = \;(1\; + \;\beta _{3} \Delta N)\; - \;\beta _{3} \Delta N\frac{{\partial \alpha _{0} ({\bf x},\;N\; + \;1)}}{{\partial \alpha _{0} ({\bf x},\;N)}}.$$

(46)

The partial derivate $\frac{{\partial \alpha_{0} ({\bf x},\;N + \;1)}}{{\partial \alpha_{0} ({\bf x},\;N)}}$ corresponds to the variation of internal variable at the end of the cycle N with respect to the variation at the beginning. It can be obtained by applying the backward Euler scheme to the internal variable evolution law $\dot{\alpha }(\tau )$ as

$${A}_{n+1}=\frac{\partial {\alpha }_{n+1}(N)}{\partial {\alpha }_{n}(N)}={\left[1-{\left(\frac{\partial \dot{\alpha }}{\partial \alpha }\right)}_{n+1}\Delta {\tau }_{n+1}\right]}^{-1}$$

(47)

Applying the chain rule to Eq. 47 recursively, Eq. 46 can be determined using the result

$$\frac{\partial {\alpha }_{0}({\bf x},N+1)}{\partial {\alpha }_{0}({\bf x}, N)}=\prod_{k=0}^{{N}_{p}}{A}_{k}$$

(48)

where N_p is the number of sampling points in the fine time scale response.

3.3 The WATMUS Method for Three-Dimensional Crack Propagation

In a finite element simulation with a single time scale method, the displacement field is the unknown to be solved. However, applying the WATMUS method, the wavelet coefficients for a complete cycle become the new variables. The iterative update of wavelet coefficients can be achieved by transforming the residual and the unknowns into the corresponding wavelet coefficients. Discretizing and linearizing the weak form of the balance of momentum using the implicit Newmark method yields the equations to compute the update of the unknowns

$${{\bf R}}^{\alpha }\left(t\right)={{\bf K}}^{\alpha \beta }\left(t\right)\Delta {{\bf u}}^{\beta }(t)$$

(49)

This equation can be expressed by index notation with a two-scale representation

$${R}_{i}^{\alpha }\left(N,{\tau }_{l}\right)={K}_{ij}^{\alpha \beta }\left(N,{\tau }_{l}\right)\Delta {u}_{j}^{\beta }(N, {\tau }_{l})$$

(50)

where R_i^α(N, τ_l) is the i-th component of the residual vector corresponding to the node α at the fine time point τ_l within the cycle N, and u_j^β(N, τ_l) represents the j-th component of the nodal displacement of the node β. Each component of the residuum and nodal displacement can be transformed into its corresponding wavelet coefficients by the DWT as described in Eq. 42

$${R}_{i,k}^{\alpha }\left(N\right)={{T}_{kl}R}_{i}^{\alpha }(N, {\tau }_{l})$$

(51)

$${u}_{j,k}^{\beta }\left(N\right)={{T}_{kl}u}_{j}^{\beta }(N, {\tau }_{l})$$

(52)

By applying the inverse DWT to Eq. 52, the iterative update of nodal displacement wavelet coefficients can be obtained

$${\Delta u}_{j}^{\beta }\left(N,{\tau }_{l}\right)={T}_{lm}\Delta {u}_{j,m}^{\beta }(N)$$

(53)

Replacing the R_i^α(N, τ_l) in Eq. 51 by Eq. 50 and substituting into Eq. 52, the corresponding transformed equations are obtained

$${R}_{i,k}^{\alpha }\left(N\right)={{T}_{kl}K}_{ij}^{\alpha \beta }\left(N, {\tau }_{l}\right){T}_{lm}\Delta {u}_{j,m}^{\beta }\left(N\right)$$

(54)

from which the update of nodal displacement wavelet coefficients are solved. The nodal wavelet coefficients for cycle N are initialized as the wavelet coefficients of the previous WATMUS cycle N–∆N as defined in Eq. 44. For the i-th iteration step in the Newmark-Beta scheme, the sets of wavelet coefficients are computed by

$$\Delta {u}_{j,m}^{\beta }\left(N\right)=-{K}_{ijkm}^{\alpha \beta }{\left(N\right)}^{-1}{R}_{i,k}^{\alpha }\left(N\right)$$

$${u}_{j,m}^{\beta }\left(N\right)={u}_{j,m}^{\beta }\left(N\right)+\Delta {u}_{j,m}^{\beta }(N)$$

(55)

with the definition of the WATMUS stiffness matrix as

$${K}_{ijkm}^{\alpha \beta }\left(N\right)={T}_{kl}{K}_{ij}^{\alpha \beta }\left(N,{\tau }_{l}\right){T}_{lm}$$

(56)

3.4 Numerical Example of Fatigue Damage Prediction

The WATMUS method is applied as an efficient and accurate method to predict the fatigue damage evolution in a three-dimensional mode-I fracture problem. A single- edge notched specimen shown in Fig. 3 is fixed to the ground and subjected to an axial cyclic load f (t) = 10.0 sin (2.5132 10⁵t) N with a period of T_s = 2.5 x 10⁻⁵ s. The shape of the structure is 10 mm x 10 mm x 1.25 mm and is discretized with 3017 structured ten-node tetrahedral elements with quadratic shape functions. The material parameters are given in Table 1. Totally 450 cycles are computed so that the computation using single time scale method can be performed within a reasonable time. The damage evolution computed by the single time scale method is compared with results obtained by the WATMUS algorithm in Fig. 4. A speed up of 15–50 times can be achieved. From the comparison it can be concluded that the WATMUS method can accelerate the prediction of fatigue damage with a quite good accuracy.

Fig. 3

A single-edge-notch block subjected to cyclic loading

Table 1 Material parameters of the 3D block

Fig. 4

Comparison of the predicted coarse scale damage evolution by the WATMUS method and the single time scale method

4 Conclusions

The application of a multi-time scaling method is inspired by the high computational cost due to the huge number of time steps for fatigue life prediction using a conventional single time scale FE model. This paper applies the wavelet transformation based multi-time scaling method (WATMUS) for predicting the fatigue damage growth for a huge number of loading cycles. Once the critical value of damage is reached, the crack propagation can be handled by a combination of the XFEM and level set methods. A fatigue damage evolution law, which is suitable for high-cycle fatigue growth is integrated efficiently by employing the WATMUS procedure. The highly oscillating response of the damage variable is retained through the wavelet coefficients obtained by a DWT, and the low frequency damage variable is transformed into a cycle scale variable. The coarse scale damage evolution is performed using the implicit second order backward difference scheme used within the WATMUS method. The performance of the WATMUS method has been investigated by applying it to a three-dimensional mode I problem with predefined planar cracks under cyclic loads. A speed up of 15–50 times can be achieved. The coupling of the XFEM and the WATMUS method can be a powerful tool for modeling three-dimensional fatigue crack growth. An entire fatigue life estimation due to crack initiation and propagation can be conducted in a reasonable computational time.