# Diagonally implicit Runge–Kutta (DIRK) integration applied to finite strain crystal plasticity modeling

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## Abstract

Diagonally implicit Runge–Kutta methods (DIRK) are evaluated and compared to standard solution procedures for finite strain crystal plasticity boundary value problems. The structure of the DIRK implementation is similar to that of a conventional implicit backward Euler scheme. It is shown that only very small modifications are required in order to transform the numerical scheme from one into the other. This similarity permits efficient adaption of the integration procedure to a particular problem. To enforce plastic incompressibility, different projection techniques are evaluated. Rate dependent crystal plasticity, using a single crystal is simulated under various load cases as well as a larger polycrystalline sample. It is shown that the two-stage DIRK scheme combined with a step size control and a time continuous projection technique for the update of the plastic deformation gradient is in general more accurate than the implicit backward Euler and an update based on exponential mapping.

## Keywords

Crystal plasticity Numerical integration Implicit Euler Diagonally implicit Runge–Kutta## 1 Introduction

A key component in modeling of crystal plasticity is the evolution equations for the plastic slip, which can be formulated as either rate independent or rate dependent, cf. [1, 2]. The rate independent assumption requires continuous identification of the active set of slip systems and can lead to an ill-conditioned problem. To avoid non-uniqueness related to the selection of the active slip systems, rate dependent plasticity is frequently used, cf. [3, 4, 5, 6]. One possibility to model rate dependent crystal plasticity models is to model the slip by a power law flow rule where all slip systems are active [7]. The drawback of the rate dependent approach used in the quasi-rate independent region is that the evolution equations become stiff. Another possibility to model rate dependence is to make use of an elastic-viscoplastic model, where the elastic domain is defined by introducing a threshold into the governing equations of the slip [8]. This approach can however lead to stiff behavior and large computational times, see [2]. In order to model polycrystals, a large number of individual grains must be considered at significant computational cost. It is therefore of high relevance to find robust and efficient numerical algorithms for solving any type crystal plasticity model.

Different numerical methods used to solve crystal plasticity models are reviewed in [6] where uniaxial loading of a single crystal is considered. It is found that a fourth order Runge–Kutta algorithm used to solve an explicit slip rate formulation is computationally more efficient than an implicit slip rate approach and an implicit deformation gradient approach, where the deformation gradient is taken as the primary unknown. To simulate instability phenomena such as shear band localization in a rate dependent viscoplastic single crystal model, an implicit numerical algorithm is used in [4] where it is shown that the integration of the stiff constitutive equations can be performed using a simple and reliable algorithm. In [9], a second order Runge–Kutta algorithm is utilized to solve a three dimensional rate-dependent crystal plasticity model and it is shown that the Runge–Kutta algorithm provides higher order accuracy than first-order explicit and implicit schemes. Another method, discussed in [10], makes use of a homotopy continuation method combined with a Newton–Raphson algorithm to solve a rate dependent crystal plasticity model. The numerical results show that the algorithm is both stable and accurate. The method of variational constitutive updates, where the internal variables are updated by means of a minimization problem can also be used for crystal plasticity models, cf. [11, 12]. It is shown in [12] that the framework of variational constitutive updates allows for analysis of a broad spectrum of engineering applications such as size effects and texture evolution in polycrystals. In [13], it is shown that using decoupling between the grains allows the response of the individual crystal to be updated in parallel which opens up for the possibility to use GPUs.

Another solution procedure is based on the idea of considering both the balance laws and the constitutive evolution equations as a system of differential algebraic equations (DAE). This approach is proposed in [14] where the DIRK scheme is used together with a multilevel Newton algorithm, cf. e.g. [15], to integrate a rate-dependent Chaboche viscoplasticity constitutive model. In [16], the DIRK scheme is used to analyze the behavior of viscoelastic elastomers and in [17] it is applied to a finite strain continuum damage model. In these studies, results are compared to the implicit backward Euler of lower order and it is shown that the embedded DIRK method increases the accuracy of the numerical solution significantly. It is also shown that the DIRK scheme preserves the conventional implicit finite element structure found when the constitutive equations are integrated using the implicit backward Euler scheme. This means that adapting a conventional multilevel finite element code to the DIRK structure involves virtually no additional implementation costs. In [18], comparisons are performed between DIRK and linear implicit Runge–Kutta and half explicit Runge–Kutta in nonlinear finite element analyses. It is concluded that the DIRK approach is the preferred approach due to its efficiency and stability.

Use of either an implicit backward Euler approach or the DIRK method in rate dependent crystal plasticity will not per se guarantee that the plastic incompressibility requirement is fulfilled. This is due to the additive structure of the integration steps. To resolve this, an exponential map is frequently used in the update of the plastic deformation gradient, cf. [19]. As an alternative to the incompressibility preserving exponential map, projection methods can be used. In [20], the inelastic incompressibility in an elastic-viscoplastic constitutive model is considered as a side condition and is solved by a projection method. The projection method can be applied at the end of each time-step or it can be solved continuously in time. Both approaches will be studied in the present work.

In this work, numerical crystal plasticity simulations of fcc crystals are performed. The numerical results obtained using the DIRK scheme are compared to the results obtained when considering an implicit backward Euler method and an exponential map along with the Padé approximation. Different projection methods are evaluated in order to enforce plastic incompressibillity. In addition, to control the time step and to increase the robustness of the solution method, an embedded DIRK scheme is evaluated.

The paper is organized as follows: the governing equations and the constitutive model are described in Sect. 2. The formulation of the numerical method is given in Sect. 3 and in Sect. 4, numerical examples related to both single crystals and a polycrystalline structure are shown. Finally, some concluding remarks close the paper in Sect. 5.

## 2 Governing equations

For the sake of completeness, the key components of the adopted crystal plasticity model are briefly presented below along with the establishment of the mechanical balance laws. For further details on the specific crystal plasticity model, the reader is referred to [19].

### 2.1 Kinematics and thermodynamics

*t*by the unique mapping \(\varvec{x} = \varvec{\varphi }(\varvec{X}, t)\). The deformation gradient is given by \(\varvec{F}=\partial _{\varvec{X}} \varvec{\varphi }\), and the right Cauchy–Green deformation tensor is defined as \(\varvec{C}= \varvec{F}^T \varvec{F}\). A multiplicative split of the deformation gradient allows \(\varvec{F}\) to be separated into an elastic and a plastic part as

*e*and

*p*are introduced to identify the elastic and the plastic components, respectively. The volume change associated with the deformation is described by \(J = \text {det} (\varvec{F})\), where \(\text {det}(\cdot )\) denotes the determinant. Since metal plasticity is considered, it is assumed that the volume change is purely elastic, i.e. plastic incompressibility is enforced, whereby it holds that \(J =\text {det}(\varvec{F}^e)\).

### 2.2 Constitutive model

*K*and

*G*are the bulk and shear moduli in the limit of small strains. The trace of a tensorial quantity is denoted by \(\text {tr}(\cdot )\). The internal variable \(g^{\alpha }\) associated with hardening on slip system \(\alpha \) enters the plastic part of the Helmholtz free energy as

*Q*is a material parameter and \(h_{\alpha \beta }\) is a hardening matrix chosen as \( h_{\alpha \beta } = \delta _{\alpha \beta } + q(1-\delta _{\alpha \beta })\). The parameter

*q*governs the ratio between the hardening of the current slip system and the hardening due to interaction with neighboring slip systems. The slip resistance \(G^{\alpha }\) that enters the dissipation inequality (11) can be identified as

*m*are material parameters which define the reference shear rate and the rate sensitivity, respectively. The material parameter \(G_0\) describes the lattice friction. For a high value of the rate sensitivity parameter

*m*, a rate-independent response is approached which results in that the resolved shear stress on slip system \(\alpha \) has to exceed the slip resistance, i.e. \(G_0 + G^{\alpha }\), in order for the slip system to become active.

*B*is a material parameter defining the saturation of the hardening. Insertion of (16) and (17) into the dissipation inequality (11) allows the dissipation inequality to be reduced to

### 2.3 Balance of linear momentum

## 3 Numerical solution procedure

### 3.1 Diagonally implicit Runge–Kutta method

*n*time steps, \({\varDelta }t_n\), and at each time step the solution \(\varvec{y}(t_{n+1})\) is approximated using

*s*stages as

*i*, respectively. At time step

*n*and stage

*i*the stage values \(\varvec{y}(t_n+c_i {\varDelta }t_n)\) are required and they are approximated in a similar manner to provide

*S*, denotes the start values which are known at time step

*n*and stage

*i*and which are obtained by evaluating

*n*and stage

*i*, respectively. The nonlinear system in (32) is solved using a multilevel Newton–Raphson scheme in each time step

*n*and at each stage

*i*.

*q*at time step \(n+1\) by \(\varvec{y}_{n+1}\) and the solution of order \(q-1\) at the same time step by \(\hat{\varvec{y}}_{n+1}\), the solution \(\hat{\varvec{y}}_{n+1}\) can be estimated using an embedded error control [14]. The embedded DIRK scheme makes use of the same weight factors \(a_{ij}\) and \(c_i\) but different \(b_i\), denoted by \(\hat{b}_i\). Thus, the solution of order \(q-1\) at time step \(n+1\) is obtained as

*q*in (28) with the solution of order \(q-1\) in (36) it is evident that the solutions are both obtained from the stage derivatives \(\varvec{f} (\varvec{Y}_{ni}, T_{ni})\), thus the local error is provided at virtually no extra cost using the embedded DIRK method.

*l*, \(z^{\xi }\) are the internal variables at Gauss point \(\xi \), \(\epsilon _r\) is a relative tolerance and \(\epsilon _u\) and \(\epsilon _z\) are absolute tolerances. Based on the error estimate, the new time step can be estimated as

(a) General Butcher tableau for the diagonally implicit Runge–Kutta method. (b) Butcher tableau for the 2-stage diagonally implicit Runge–Kutta method, see [14]. (c) Butcher tableau for the backward Euler method

(a) | (b) | (c) | |
---|---|---|---|

\(\begin{array}{c|cccc} c_1 &{} a_{11} &{} 0 &{} \cdots &{} 0 \\ c_2 &{} a_{21} &{} a_{22} &{} \cdots &{} 0 \\ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ c_s &{} b_{1} &{} b_{2} &{} \cdots &{} b_{s} \\ \hline &{} b_{1} &{} b_{2} &{} \cdots &{} b_{s} \\ &{} \hat{b}_{1} &{} \hat{b}_{2} &{} \cdots &{} \hat{b}_{s} \end{array}\) | \(\begin{array}{c|cc} \alpha &{} \alpha &{} 0 \\ 1 &{} 1-\alpha &{} \alpha \\ \hline &{} 1-\alpha &{} \alpha \\ &{} 1-\hat{\alpha } &{}\hat{\alpha } \end{array}\) | \(\begin{array}{l} \alpha = 1-\frac{1}{2} \sqrt{2} \\ \hat{\alpha } = 2-\frac{5}{4} \sqrt{2} \\ \end{array}\) | \(\begin{array}{c|c} 1 &{} 1 \\ \hline &{} 1 \\ \end{array}\) |

### 3.2 Implicit backward Euler method

### 3.3 Exponential update

### 3.4 Plastic incompressibility

As mentioned previously, for a model in which small strains are assumed, neither the exponential update nor the projection method is needed and the plastic incompressibility is fulfilled without any extra computational costs.

## 4 Numerical examples

Material parameters used in the constitutive model, representative for austenitic steel [19]

Parameter | | | | \(G_0\) | | \(\dot{\gamma }_0\) | | | \(g^{\alpha }_0\) |
---|---|---|---|---|---|---|---|---|---|

Value | 80 MPa | 164 MPa | 200 MPa | 55 MPa | 1.4 | \(10^{-3}\) 1/s | 50 | 8 | \( 7\times 10^{-3}\) |

Numerical parameters for the adaptive DIRK method

Parameter | \(\epsilon _r\) | \(\epsilon _u\) | \(\epsilon _{F^p}\) | \(\epsilon _g\) | \(f_{max}\) | \(f_{min}\) | \(f_s\) |
---|---|---|---|---|---|---|---|

Value | \(10^{-9}\) | \(2 \times 10^{-3}\) | \(2.7 \times 10^{-3}\) | \(1.8 \times 10^{-3}\) | 2 | 0.5 | 0.9 |

*N*is the number of grains in each Gauss point. Results will be shown for the internal variables by finding the maximum hardening variable \(g^{\alpha }\) in all slip systems. The maximum component of \(\varvec{F}^p\) will also be studied. To simplify the notation, the following definitions will be used subsequently

### 4.1 Uniaxial tensile deformation of a one-element single crystal

The von Mises effective stress is plotted in Fig. 2 where a maximum effective stress slightly above 800 MPa is obtained at a displacement of \(u_x=3\) mm. In Fig. 3, the relative errors of the von Mises effective stress, \(g_{max}\) and \(F^p_{max}\) are shown for the different methods. Both the BE and the DIRK schemes simulated with the projection method in (46)–(53) result in significant errors in the von Mises effective stress and in the relative error of \(F^p_{max}\). The relative errors in the stresses and in \(F^p_{max}\) obtained from BE are the highest whereas the DIRK scheme performs slightly better. Considering the relative error of \(g_{max}\) shown in Fig. 3b, the BE scheme generates the largest relative error.

It can be concluded that the BE scheme is systematically less accurate compared to the DIRK and EU schemes and that use of a time continuous projection technique will improve the results in terms of a lower relative error, at least when considering the effective stress and the plastic deformation gradient. Based on this conclusion, only results from using the time continuous projection technique will be considered from here on.

Effective von Mises stress obtained from the reference simulation, i.e. the EU scheme with a time step \({\varDelta }t = 1\times 10^{-6}\) s

**a**The relative error of the von Mises effective stress,

**b**the relative error of \(g_{max}\) and

**c**the relative error of \(F^p_{max}\) found when using EU

### 4.2 Simple shear deformation of a single crystal

Next, simple shear deformation of a single crystal is modeled, again using a single 3D 8-node element of dimensions \(1 \times 1 \times 1 \ \text {mm}^3\) , cf. Fig. 1c. The initial orientation of the crystal is defined by the Euler angle set \((\phi _1,{\varPhi },\phi _2) =(0,0,0)\) and the BE and EU schemes are used with the time step \({\varDelta }t=0.4\) s. The simulations are performed with a constant displacement rate of \(\dot{u}_x = 0.1\) mm/s, applied as shown in Fig. 1c for a total time of 30 s.

As discussed in [12, 23], the crystal orientation will, in this deformation mode, continuously change. Figure 5 shows that the von Mises stress is continuously changing due to the evolution of the crystal orientation and as a result the relative errors in the stresses shown in Fig. 6a are significantly higher compared to the uniaxial load case. The peaks in the errors can be explained by additional slip systems being activated as deformation progresses.

*e*from the embedded DIRK, cf. (38), is plotted together with the time step versus the displacement in Fig. 7. The fluctuations in the error and the time step are due to activation of different slip systems.

*e*

*e*, see (38)

**a** Illustration of the plate with a hole subjected to uniaxial deformation. **b** Finite element discretization of one eighth of the plate where advantage is taken of the symmetry of the geometry in **a**

As a conclusion from the single crystal examples, the DIRK scheme with the time continuous projection technique and the EU scheme are the two integration methods which generate the smallest relative errors compared to the other methods.

### 4.3 Uniaxial tension of a plate with hole

Effective von Mises stress obtained from the reference simulation, i.e. the EU scheme with a time step \({\varDelta }t = 1\times 10^{-6}\) s

**a**The relative error of the von Mises effective stress,

**b**the relative error of \(g_{max}\) and

**c**the relative error of \(F^p_{max}\) found when using EU

As a final example, a polycrystalline plate with a centrally located hole is considered, cf. Fig. 8. The plate has the dimensions \(100 \times 100 \times 10\,\text {mm}^3\) and due to symmetry only one eighth of the structure is modeled. The finite element mesh consists of 370 brick elements and the plate is subjected to tensile deformation with a constant displacement rate \(\dot{u}_x = 5\) mm/s for a total time of 1 s. Each Gauss point comprises \(N= 100\) randomly oriented grains. The rate sensitivity parameter *m*, cf. (16), is chosen as \(m=20\), which is lower than the value in Table 2, to decrease the computational time. The time step used in the BE and EU schemes is \({\varDelta }t = 0.004\) s.

*e*

In Fig. 11, the error *e* from the embedded DIRK method, cf. (38), is plotted together with the time step \({\varDelta }t\). The error is fluctuating around or slightly below 1 and thus the time step changes are negligible.

Summarizing the results from the single and polycrystal examples, the DIRK method provides higher accuracy when combined with the time continuous projection technique. In addition, the embedded DIRK method offers a time step control at no additional computational costs. It can be concluded that DIRK integration method should be the method of choice when solving crystal plasticity boundary value problems.

## 5 Conclusion

The implications of using a diagonally implicit Runge–Kutta (DIRK) scheme in crystal plasticity simulations have been scrutinized. The DIRK approach is compared to a classical implicit backward Euler (BE) scheme and an exponential update procedure. Different projection techniques have been evaluated to enforce plastic incompressibility. For the DIRK scheme, an embedded error estimate is used to control the step size. Both single crystal and polycrystal simulation examples are considered

The relative error in terms of stresses, hardening variables and plastic deformation are investigated. It is shown that the choice of projection method to ensure plastic incompressibility is crucial for the DIRK and the implicit backward Euler schemes. A time-continuous projection approach is shown to be able to enforce plastic incompressibility without reducing the accuracy.

The single crystal simulations show that the exponential update procedure generates relative errors of the same order as the relative error from the implicit backward Euler approach, while the DIRK scheme provides higher accuracy. It is also shown that in polycrystalline simulations, the DIRK approach provides significantly higher accuracy than the methods used for comparison in the present work.

In conclusion, since the DIRK scheme results in higher accuracy with negligible additional cost, the DIRK scheme appears to be the preferable time integration scheme for crystal plasticity, when combined with a time-continuous projection approach.

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