Daniele Picone

Mohamad Jouni

Mauro Dalla Mura

SPECTRO-SPATIAL HYPERSPECTRAL IMAGE RECONSTRUCTION FROM INTERFEROMETRIC ACQUISITIONS

Keywords: Optimization, spectrometers, hyperspectral imaging, image reconstruction

In the last decade, novel hyperspectral cameras have been developed with particularly desirable characteristics of compactness and short acquisition time, retaining their potential to obtain spectral/spatial resolution competitive with respect to traditional cameras. However, a computational effort is required to recover an interpretable data cube. In this work we focus our attention on imaging spectrometers based on interferometry, for which the raw acquisition is an image whose spectral component is expressed as an interferogram. Previous works have focused on the inversion of such acquisition on a pixel-by-pixel basis within a Bayesian framework, leaving behind critical information on the spatial structure of the image data cube. In this work, we address this problem by integrating a spatial regularization for image reconstruction, showing that the combination of spectral and spatial regularizers leads to enhanced performances with respect to the pixelwise case. We compare our results with Plug-and-Play techniques, as its strategy to inject a set of denoisers from the literature can be implemented seamlessly with our physicsbased formulation of the optimization problem.

INTRODUCTION

Characterizing the spectrum of a light source (i.e., measuring its intensity at each wavelength) is at the core of imaging spectroscopy, and has deep implications in various fields, such as geology, gas detection, security, remote sensing, disaster prevention, and more [START_REF] Michael | Hyperspectral remote sensing[END_REF][START_REF] Manolakis | Hyperspectral imaging remote sensing: physics, sensors, and algorithms[END_REF]. In recent times, both the scientific community and industrial venues have shown interest in hyperspectral image (HSI) spectrometers which operate on the principle of interferometry, as they potentially allow for Mauro Dalla Mura is also with Institut Universitaire de France (IUF), 75005 Paris, France. This work is partly supported by grant ANR FuMulti-SPOC (ANR-20-ASTR-0006), and partly by Région Auvergne-Rhône-Alpes grant "Pack Ambition International 2021" (21-007356-01FONC, 21-007356-02INV).

instruments with reduced cost and dimensions and for acquisitions with finer spectral resolution and improved SNR [START_REF] Hecht | Optics[END_REF][START_REF] Hariharan | Basics of interferometry[END_REF].

Hyperspectral cameras based on interferometry measure the interference of superimposed coherent light waves which travel across different optical paths, known as the optical path differences (OPDs). At the detector level, a grayscale image is formed for each value of the OPD. The technique of capturing data across different OPDs is called Fourier-transform spectroscopy (FTS). Finally, a datacube is obtained where each pixel of the observed scene a vector of values in the domain of OPD known as the interferogram.

However, in order to obtain the HSI, two issues arise. First, a data processing step is required to reconstruct the spectral response in the domain of the wavelengths, as the acquisitions are only available in a transformed domain. Second, the spatial information should be taken into account in the reconstruction as a pixel-to-pixel spectral reconstruction [START_REF] Jouni | Model-based spectral reconstruction of interferometric acquisitions[END_REF] ignores spatial noise and pixel neighborhood information. Given the nature of this computational imaging [START_REF]Computational methods for inverse problems in imaging[END_REF] problem, a combination of spectral and spatial regularizers is required.

Machine learning-based approaches may provide a way to intrinsically regularize this formulation, but the availability of training data is limited in our scenario. As we deal with prototype instruments, it is complicated to build a large enough database of coupled acquisitions and references. To partly address this issue, hybrid (model-and learning-based) approaches and basic Plug-and-Play (PnP) approaches [START_REF] Ulugbek S Kamilov | Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications[END_REF], such as algorithm unrolling [START_REF] Vishal Monga | Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing[END_REF] and deep priors [START_REF] Zhang | Plug-and-play image restoration with deep denoiser prior[END_REF], are currently gaining traction in the community. They rely heavily on a good understanding of the system, which we aim to address here as preliminary step and solid basis for the interpretability of such systems.

In this work, we propose a spectro-spatial model-based inversion for the HSI reconstruction of interferometric image acquisitions, by exploiting the knowledge of the physics and the characteristics of interferometric systems and by showing the importance of having priors that are adapted to the data. This is an extension to our previous work [START_REF] Jouni | Model-based spectral reconstruction of interferometric acquisitions[END_REF] which was only pixel-to-pixel based.

In this paper, we extend the study to combine spatial reg-ularizers with the spectral ones thanks to collaborative total variation (CTV) [START_REF] Duran | Collaborative total variation: a general framework for vectorial tv models[END_REF], applied on the case where a potentially infinite amount of interfering waves can be superimposed. For example, this is capable to also deal with several instruments based on Fabry-Pérot (FP) interferometers [START_REF] Zucco | A hyperspectral imager based on a fabry-perot interferometer with dielectric mirrors[END_REF][START_REF] Oiknine | Multiaperture snapshot compressive hyperspectral camera[END_REF][START_REF] Pisani | Compact imaging spectrometer combining fourier transform spectroscopy with a fabry-perot interferometer[END_REF][START_REF] Picone | Interferometer response characterization algorithm for multi-aperture fabry-perot imaging spectrometers[END_REF]; this is defined in our work as ∞-wave model, also known as Airy distribution [START_REF] Hecht | Optics[END_REF]. We propose to solve the inversion problem through a simple enough solver, notably the Loris-Verhoeven (LV) algorithm [START_REF] Loris | On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty[END_REF], with proximal operators that better characterize the a priori knowledge through induced sparsity on a spectro-spatial cascaded Fourier domain transform and total variation (TV) operations. We compare our results with pixel-to-pixel spectral reconstruction [START_REF] Jouni | Model-based spectral reconstruction of interferometric acquisitions[END_REF] and PnP techniques, especially when the acquisitions are affected by noise. Our contributions can be summed up as follows: (i) we propose a model-based HSI reconstruction method from interferograms based on spectro-spatial regularization, (ii) we perform a comparison with image priors based on PnP techniques within the context of imaging spectrometers based on interferometry, to show the importance of having priors that are adapted to the data.

PROPOSED METHODOLOGY

Problem statement

We want to set up the reconstruction problem as a Bayesian problem. Let X ∈ R I×J×K describe an unknown datacube to reconstruct representing the scene with I × J pixels and K channels. The acquisition process is then described as a stochastic process in the form:

Y = A(X ) + E . (1)
In the above equation, E denotes additive white Gaussian noise and Y ∈ R I×J×L describes our acquisition. The latter retains the same spatial resolution of our target datacube, while its spectral content is defined in a transformed domain with L samples. This domain transformation is defined by a linear operator A, which we define as:

A(X ) = A • 3 X , (2)
where • 3 denotes a contraction operator along the third modality, i.e., a matrix multiplication along the third modality of the HSI tensor X , and A ∈ R L×K is the transmittance response of our acquisition system. The goal of the inversion protocol is to estimate the value of X . We propose to derive an estimation X of the spectrum X by setting up a Bayesian problem [START_REF] Idier | Bayesian approach to inverse problems[END_REF] with spectro-spatial regularization. We specifically aim to minimize a cost function in the form:

X = argmin X 1 2 ∥Y -A(X )∥ 2 2 + λg(LX ) , (3)
where the right hand side is expressed as the sum of a data fidelity term and a term which characterizes the a priori knowledge on the spectrum. In the above equation, ∥ • ∥ 2 defines the ℓ 2 norm applied over the concatenation of all pixels of its argument, while g(•), L, and λ > 0 denote a scalar functional, a linear operator and a regularization parameter, respectively.

In the following sections we will detail the direct transformation and the regularization term.

Interferometer transmittance function

The interferometric acquisition system is characterized by a given transmittance response A. In the context of interferometry, which is the focus of our work, this defines a transformation from the spectral domain to an interferogram. For FTSs, it is common practice to express the spectral domain in terms of wavenumbers, i.e., the reciprocal of wavelengths. This spectrum is continuous in nature, but can be represented with a sufficiently fine sampled discretization of the wavenumber range {σ k } k∈[1,...,K] . On the contrary, the set of OPDs {δ l } l∈[1,...,L] where the interferogram is sampled is limited by the capability of the instrument of generating L different optical paths. While many techniques are available to measure an interferogram from a spectrum [START_REF] Pisani | Compact imaging spectrometer combining fourier transform spectroscopy with a fabry-perot interferometer[END_REF][START_REF] Picone | Interferometer response characterization algorithm for multi-aperture fabry-perot imaging spectrometers[END_REF][START_REF] Hariharan | Optical interferometry[END_REF], we only remind here that for Fabry-Perot interferometers the coefficients a ik of the transmittance response A can be modeled with an Airy distribution [START_REF] Hecht | Optics[END_REF][START_REF] Picone | Interferometer response characterization algorithm for multi-aperture fabry-perot imaging spectrometers[END_REF]:

a lk = (1 -R) 2 1 + R 2 -2R cos(2πσ k δ l ) , (4)
where R denotes the reflectivity of the surface of the interferometric cavities.

Proposed reconstruction algorithm

In this work, we propose to describe the linear operator L(X ) = L (TV) (L (DCT) (X )) from eq. ( 3) as the cascade of two operators L (DCT) and L (TV) , which respectively operate in the spectral and spatial domain. We define them as follows:

1. Spectral operator L (DCT) : we define it as type-II discrete cosine transform (DCT). Specifically, we define a square matrix W ∈ R K×K , whose coefficients are:

w ij = 2 K cos π K j - 1 2 (i -1)
∀i,j∈{1,...,K} .

(5) The matrix is applied to the third modality of the latent HSI X and we denote the result of this operation by Z = L (DCT) (X ) = W • 3 X . The choice is taken as a follow-up of the results of our previous work [START_REF] Jouni | Model-based spectral reconstruction of interferometric acquisitions[END_REF], as the DCT has been shown to be particularly effective to induce sparsity on the spectrum.

2. Spatial operator L (TV) : we define it as TV using the formulation of [START_REF] Condat | Discrete total variation: New definition and minimization[END_REF]. The operator is applied on the spatial modalities of Z and returns a tensor T ∈ R I×J×K×2 such that ∀ i, j, k:

T i, j, k, 1 = Z i+1, j, k -Z i, j, k , T i, j, k, 2 = Z i, j+1, k -Z i, j, k . (6)
In other words, this is the concatenation of the horizontal (columns) and the vertical (rows) differential operators, along a newly introduced fourth modality.

For the regularization term g(•), we propose the collaborative norm ℓ 1,1,1,2 , i.e., by imposing the ℓ 2 norm on the fourth modality of T and the ℓ 1 norm on the remaining three modalities. This procedure, known as CTV, was originally proposed in [START_REF] Duran | Collaborative total variation: a general framework for vectorial tv models[END_REF] to combine different modalities, although their original formulation did not include any spectral operator.

The problem is then solved by primal-dual splitting algorithm known as Loris-Verhoeven [START_REF] Loris | On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty[END_REF] with over-relaxation [START_REF] Condat | Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists[END_REF], aiming to minimize the fidelity term of eq. ( 3), where we plugged the DCT and TV priors in a collaborative environment. Algorithm 1 describes the iterative updates of our proposed method. A T and L T represent the adjoint operators of A and L, respectively, while ∥A∥ op and ∥L∥ op represent their operator norms [START_REF] Condat | Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists[END_REF]. X and U are the primal and dual variables, respectively. In our context, prox λ,g ⋆ (T ) defines the proximal operator associated to the Fenchel conjugate of the ℓ 1,1,1,2 norm [START_REF] Parikh | Proximal algorithms[END_REF]. ∀ i, j, k, and denoting by T i, j, k, : the (i, j, k)-th vector of T of dimension 2 (along the fourth modality), this operator becomes:

prox λ,g ⋆ (T i,j,k,: ) = λ T i,j,k,: ∥T i,j,k,: ∥2 if ∥T i,j,k,: ∥ 2 > λ T i,j,k,: if ∥T i,j,k,: ∥ 2 ≤ λ (7)
The values of η, τ , and ρ were chosen according to the relevant literature to ensure convergence [START_REF] Condat | Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists[END_REF].

EXPERIMENTS AND RESULTS

Experimental setup

To test the effectiveness of the reconstruction algorithms, we operate in a simulated scenario. Specifically, we performed our tests using a HSI that we acquired with the commercial hyperspectral camera Specim IQ, whose scene contains a colorchecker and some geometrical targets. To generate our reference, we spatially resized the raw acquisition to 96 × 96 pixels and interpolated the channels to regularly sample the range of visible wavenumbers. This range is between [1.4, 2.5] µm -1 and contains 366 channels. We then generated a transmittance response for our interferometric camera using eq. ( 4), using parameters replicating the setup of Pisani and Zucco in [START_REF] Pisani | Compact imaging spectrometer combining fourier transform spectroscopy with a fabry-perot interferometer[END_REF]. We made this choice as their setup is already optimized for reconstructing spectra in the visible Algorithm 1 Proposed method by Loris-Verhoeven [START_REF] Loris | On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty[END_REF] Require: A, L, N iters Initialize X (0) = A T Y, U (0) = LX (0) Initialize τ =0.99/∥A∥ 2 op , η=1/(τ ∥L∥ 2 op ), and ρ=1.9 Define prox λ g ⋆ (•) as in eq. ( 7) while Stopping criterion is not met do 3.208 0.9853 LV-DCT [START_REF] Jouni | Model-based spectral reconstruction of interferometric acquisitions[END_REF] 3.046 0.9716 LV-TV 2.737 0.9794 LV-CTV (ours)

V (q) = A T (AX (q) -Y) X (q+ 1 2 ) = X (q) -τ V (q) + L T U (q) U (q+ 1 2 ) = prox λ,g ⋆ U (q) + ηLX (q+ 1 2 ) X (q+1) = X (q) -ρτ V (q) + L T U (q+ 1 2 ) U (q+1) = U (q) + ρ U (q+ 1 2 ) -U (q) end while return X = X (Niters)
1.523 0.9946 range and we assume for simplicity that the the interferogram can be sampled even at very low OPDs. More specifically we chose the interferogram domain to be regularly sampled within an OPD range in [0, 8] µm with 25 nm step, so that each pixel is represented by 319 samples; the reflectivity is set to R = 0.255. Finally, we add Gaussian noise to the simulated acquisition obtaining a tensor Y ∈ R 96×96×319 , so that the mean-centered version of the obtained signal has a signal-to-noise ratio (SNR) of 10 dB. This signal is processed with a series of techniques that will be discussed in the next section, in order to reconstruct an image datacube X ∈ R 96×96×366 , to be compared with our same-sized reference X . The objective analysis will be performed using the structural similarity index measure (SSIM) and root mean squared error (RMSE) quality indices.

Results and discussion

The reconstruction algorithms used in our tests employ two frameworks. The first one employs our own implementation of the Loris-Verhoeven (LV) algorithm [START_REF] Loris | On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty[END_REF], in the overrelaxed version [START_REF] Condat | Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists[END_REF] that we used in a previous work [START_REF] Jouni | Model-based spectral reconstruction of interferometric acquisitions[END_REF].

In this framework we tested three different linear operators for the regularizer L to prove the effectiveness of the proposed inversion strategy described in Section 2.3. In particular, we tested our setup both using the each of the DCT operator L (DCT) and the TV operator L (TV) separately, then using them in cascade as described in Section 2.3. The three methods are denoted respectively as DCT, TV and CTV in figures and tables. We also tested our reconstruction problem employing a PnP strategy [START_REF] Singanallur | Plug-and-Play priors for model based reconstruction[END_REF]. For this purpose, eq. ( 3) was solved with the linearized alternating direction method of multipliers (ADMM) [START_REF] Yang | Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization[END_REF] implementation in SCICO [START_REF] Balke | Scientific computational imaging code (SCICO)[END_REF],

where the proximal operator associated to the reconstruction is substituted with a general purpose denoiser. We tested this strategy with both an isotropic TV and a block-matching and 3D filtering (BM3D) [START_REF] Dabov | Image restoration by sparse 3D transform-domain collaborative filtering[END_REF] denoiser. For all tested reconstruction methods the hyperparameters were adjusted to the best of our abilities and the tests were ran for 100 iterations, except for the BM3D, which was run for 10 iterations as it is much slower than the counterparts. The objective quality results, shown in Table 1, show the benefit of the joint spectral and spatial regularization, especially in the LV framework. The results are confirmed by the visual comparison and given in Figure 1 and2. The application of TV allows to strongly improve the spatial coherence of the reconstructed image. This is an expected result for the image that we used for our test, which is mostly patch based, for which the TV is known to have competitive performances. On the contrary, strong distortions are still present with PnP based algorithms. We should however note that we are operating in a disadvantageous scenario for them to be applied in their unaltered form, as BM3D is typically better suited for images with higher spatial resolution, that enhance its capability to derive common spatial features on the scene. This shows the importance of using priors for the regularizers that are adapted to the data. In general terms, the LV framework converges faster than the ADMM, as proven by the results with the TV regularizer.

CONCLUSION

In this work, we have shown the importance of the spatial regularization for the reconstruction of datacubes acquired in the domain of the interferograms. In future works, we plan to extend these results by further investigating PnP methods adapted for hyperspectral data, exploring non-redundant domain transformations, such as tensor decompositions.

Fig. 1 :

 1 Fig. 1: Reconstruction results with different inversion methods, using optimized regularization and convergence parameters. For visualization, three channels from the HSI are chosen close to the central RGB wavelengths.

Fig. 2 :

 2 Fig. 2: Visualization of the acquisition and the reconstruction for the central pixel of the green colorchecker box. (a) Comparison between noiseless and noisy acquisition with SNR of 10dB. (b) Comparison of reconstructed spectra with a selection of the inversion methods described in this paper.

Table 1 :

 1 Objective quality assessment reconstruction results.

	Method	RMSE [×10 -2 ] SSIM
	Reference	0	1
	ADMM-TV	3.101	0.9857
	ADMM-BM3D [21]


Miaoran Liu

Afia Kouadri-Henni
email: author.afia.kouadri-henni@insa-rennes.fr

Benoit Malard

SIMULATION OF LOW-CYCLE FATIGUE RESIDUAL STRESS IN DP600 STEEL LASER-WELDED STRUCTURE

Keywords: Finite element method, Residual stress, Low-cycle fatigue, Laser welding. _______________________________________________________________________________________________________________________________

With the Abaqus finite element method, the residual stress of DP600 steel after laser welding and low-cycle fatigue is simulated. Due to the large temperature gradient during laser welding and the cyclic loading during low-cycle fatigue, the material produces plastic deformation and strain hardening, in which strain is one of the main parameters affecting strain hardening. This paper first establishes plastic material constitutive models to simulate the laser welding residual stress. Then, low-cycle fatigue residual stress is simulated by the combined hardening models. Finally, the variability of residual stress is analyzed. The results show that residual stress is redistributed and relaxes after low-cycle fatigue. Furthermore, the simulation results are verified by experimental data.

Introduction

Modern welding technology first appeared at the end of the 19 th century. With the rapid development of welding technology, it can be divided into laser welding, arc welding, and electron beam welding according to different heat sources. Laser welding is a precision welding method that uses a high energy density laser beam as the heat source. Compared with arc welding and electron beam welding technology, the advantages of laser welding are well known: small heat input, fast welding speed and cooling speed, and especially due to the small heat-affected zone, the residual stress is small, etc. [START_REF] Liu | Numerical modeling and experimental verification of residual stress in autogenous laser welding of high-strength steel[END_REF] . Therefore, laser welding is widely used in various fields, such as aerospace, automobile production, and shipping.

Laser welding is a process involving rapid heating and cooling. The weld cannot expand or contract freely due to the restrictions on the temperature in the surrounding area. Residual stress and strain will be generated after cooling. In practice, the residual stress of the welded structure usually does not show up immediately. However, it will gradually play a role due to the superposition of stress during the service. Since the welded structure is often subjected to dynamic cyclic loading, when the loading is applied to the welded structure containing residual stress, the loading and the residual stress are combined, which affects the mechanical properties, especially fatigue properties [START_REF] Cho | Fatigue strength in laser welding of the lap joint[END_REF] . Generally, compressive residual stress is beneficial in prolonging the fatigue life and can sometimes be used to improve fatigue strength. However, tensile residual stress plays a detrimental role in fatigue properties, which can cause deformation and cracks. Therefore, research on the welded structure before and after lowcycle fatigue is meaningful.

Laser welding is characterized by highly collimated and concentrated laser beam energy, which makes it challenging to obtain the temperature and strain near the fusion zone during the experiment. In the traditional design process, fatigue properties are usually obtained through a large number of experiments. Not only the experimental period is long, and the cost is high, but many experimental parameters and relationships cannot be directly obtained from the experiment. The development of finite element technology allows us to obtain accurate temperature and stress field results, saving time and money. Abaqus and other numerical methods can be used to investigate the residual stress field as well as the evolution of the residual stress during the laser welding and low-cycle fatigue process.

Recently, the research on residual stress has primarily focused on the effects of process parameters on laser welding residual stress [START_REF] Chuan | Numerical and experimental analysis of residual stresses in full-penetration laser beam welding of Ti6Al4V alloy[END_REF] ; the influence of residual stress on microstructure, deformation and mechanical properties [START_REF] Rong | Laser beam welding of 316L T-joint: microstructure, microhardness, distortion, and residual stress[END_REF][START_REF] Yilbas | Laser welding of AISI 316 steel: microstructural and stress analysis[END_REF] ; due to the increased use in automotive, fatigue properties of laser-welded dual-phase steel and high-strength low-alloy steel have been a topic of investigation [START_REF] Westerbaan | Effects of concavity on tensile and fatigue properties in fibre laser welding of automotive steels[END_REF][START_REF] Xu | Tensile and fatigue properties of fiber laser welded high strength low alloy and DP980 dual-phase steel joints[END_REF][START_REF] Parkes | Microstructure and fatigue properties of fiber laser welded dissimilar joints between high strength low alloy and dual-phase steels[END_REF] .

This paper aims to explore the influencing mechanism of laser welding residual stress on low-cycle fatigue residual stress. First, laser welding residual stress is acquired by different plastic material constitutive models. The obtained residual stress is imported as the predefined stress field for the following simulation. Then, the combined hardening models are employed to evaluate the low-cycle fatigue residual stress. Finally, the effects of laser welding residual stress on low-cycle fatigue residual stress are studied. Furthermore, the simulation results are verified by experimental data.

Material

In this paper, DP600 steel is used as the base metal. Usually, DP steel is made by annealing and cooling rapidly and is one kind of AHSS (Advanced High Strength Steel). The microstructure matrix of DP600 steel is ferrite with good plasticity and toughness, and it combines with martensite to improve hardness. The ductility of steel is derived from ferrite, and the strength is derived from martensite. The microstructure of DP600 steel is shown in Fig. 1 [START_REF] Mccallum | Characterization of DP600 Steel Subject to Electrohydraulic Forming[END_REF] . DP600 steel has been considered for many applications due to its high tensile strength (654±14 MPa) and low yield strength (310±40 MPa) compared to conventional grades of steel [START_REF] Farabi | Microstructure and mechanical properties of laser welded DP600 steel joints[END_REF] .

Simulations

The entire numerical computations consist of thermal and elastic-plastic mechanical and low-cycle fatigue analyses. These two parts are related to the calculation of temperature, laser welding residual stress, and low-cycle fatigue residual stress, respectively. All computations are conducted on the Abaqus platform. This paper focuses on the effects of plastic material constitutive models on the residual stress and the residual stress before and after low-cycle fatigue. The simulation results are further verified through the experiment.

Finite element model

The specimen geometry used for the finite element model is shown in Fig. 2, where the thickness of the base metal is 1.25 mm, with a 0.1 mm gap between the two base metals.

Laser lap welding is used in this paper, with a laser power of 3500 W and a welding speed of 3 m/min. In the low-cycle fatigue simulation, the tensile loading varies a sine function of time with an amplitude of 400 MPa, a stress ratio of 0, and a frequency of 5 Hz.

During the laser welding simulation, temperature-dependent material properties such as thermal conductivity, specific heat, thermal expansion, mass density, Young's modulus and Poisson's ratio are considered. The low-cycle fatigue simulation is carried out at room temperature, the value of material properties at 25 ℃ is used.

Boundary conditions

Boundary conditions in the thermal and mechanical analysis should be defined. The room temperature of 25 ℃ is considered as the initial condition in both laser welding and low-cycle fatigue simulation. For the boundary conditions in thermal analysis, heat loss due to thermal conduction, convection and radiation is considered. The boundary conditions in the mechanical analysis not only eliminate the rigid body displacements of the model but also do not affect the free deformation of the structure, the same as the experimental conditions.

Heat source model

Different heat source models have different heat flux distributions and can lead to different heat absorption of materials, which not only affects the accuracy of the laser welding simulation but also affects the distribution of the temperature and residual stress fields. Therefore, selecting the heat source model is very important, as it needs to truly reflect the characteristics of the laser heat source. The conical heat source with Gaussian distribution is used as the heat source model in this paper. The schematic of the conical heat source with Gaussian distribution is shown in Fig. 3. This type of heat source helps simulate the welding process with high energy and obtain the weld with concentrated energy and a large aspect ratio, which is closer to the actual situation of laser self-fusion welding [START_REF] Kik | Heat source models in numerical simulations of laser welding[END_REF] . In Abaqus, the parameters of the heat source model are defined by the DFLUX subroutine.

Plastic material constitutive models

The nominal stress-strain curve can be obtained from the tensile experiment. However, the cross-sectional area and length are nonlinear during the tensile process. The nominal stressstrain curve does not reflect the constitutive relationship in the plastic period. It is necessary to obtain the true stress-strain curve. In order to continue the deformation, external stress must continue to increase when it exceeds the yield strength. This phenomenon is known as strain hardening. Since the external loading applied in the low-cycle fatigue experiment is usually equal to or higher than the yield strength, the material constantly generates strain hardening during the plastic deformation. In summary, the large temperature gradient and applied loading cause plastic deformation and strain hardening, affecting the constitutive relationship and influencing residual stress. It is necessary to establish constitutive models to study the relationship between laser welding and low-cycle fatigue residual stress.

In Abaqus, different plastic material constitutive models can be defined using the UHARD subroutine. In this paper, three strain hardening models are used to describe the plastic deformation ability at a fixed temperature and strain rate [START_REF] Chen | Validation of constitutive models for experimental stress-strain relationship of high-strength steel sheets under uniaxial tension[END_REF] during the laser welding process, namely, the Ludwik Model, the Voce Model, and the Ludwik-Voce Model. Ludwik Model is suitable to describe the strain hardening capacity of martensite combined with a strong hardening effect. And the Voce Model is used to describe material recovery, such as the plastic flow of ferrite. The Ludwik-Voce Model is better for simulating the stress change after the strain value of 0.2. It is a combination of the Ludwik Model and the Voce Model. In the Ludwik-Voce Model, ω is the proportion of martensite that changes in the welding process (0.2 in this paper) [START_REF] Ramazani | Characterization of microstructure and mechanical properties of resistance spot welded DP600 steel[END_REF] . In equations, the yield strength is shown by a stress factor (σ0), K, K1, K2, n, n1 and n2 are material parameters. The experimental data come from the stress-plastic strain of the tensile experiment. The fitting curves are shown in Fig. 4.

Ludwik Model

n 0 Kε σ σ   (1) Voce Model   ) n exp( 1 K σ σ 0     
(2)

Ludwik-Voce Model     ε) n exp( 1 K σ ω) (1 ) ε K ω(σ σ 2 2 0 n 1 0 1        (3)

Low-cycle fatigue model

Generally, the approach for determining fatigue properties of the welded structure is to establish the S-N curve. This method repeatedly applies periodic loading to the structure until a stable state is obtained. However, this technique is conservative and expensive because many loading cycles may be required before achieving a stable response. Abaqus uses the direct cyclic technique, which combines Fourier series and time integration of the nonlinear material behavior to obtain a stable response directly. The direct cyclic technique effectively solves the problem of time cost due to the need to apply many loading cycles before the stable state is obtained. The basis of this technique is to construct a displacement function (t) u to describe the response of the structure at all times t during a load cycle with period T [14] .

Results and discussions

Fig. 5 shows the longitudinal and transversal laser welding residual stress curves with three models. All models describe well the residual stress in laser welding of the welded structure. The average value of the longitudinal residual stress is about 180 MPa, the transversal residual stress is about 180 MPa in the center of the weld, and the maximum is about 210 MPa in the heat-affected zone. The residual stress in the heat-affected zone is high because when the heat source acts on a certain point of the weld during laser welding, it will cause the material to expand due to the temperature variation and form an area of compressive stress. The compressive stress is higher than the yield strength at this temperature, so the material undergoes compressive plastic deformation. When the heat source continues moving, the material gradually cools down, and the yield strength increases, thus resulting in high residual stress. Fig. 6 shows the longitudinal and transversal low-cycle fatigue residual stress curves with three models. It can be seen from Fig. 6 that because one end of the welded structure is fixed and the other end is subjected to cyclic loading, the mid-planes of the two base metals are not in a straight line. Under cyclic loading, the welded structure is subjected to torque around the weld. So the distribution of transversal low-cycle fatigue residual stress is asymmetric. The Voce Model is used to describe the plastic flow of the ferrite. Because the ferrite content of the welded structure is larger than the martensite, the Voce Model has a more significant effect on low-cycle fatigue residual stress than the Ludwik Model. The longitudinal residual stress in all models is higher on both sides and lower in the center. In low-cycle fatigue, an important part of the residual stress is the plastic stress generated by plastic deformation during cyclic loading. When only one end of the welded structure is subjected to cyclic loading, the deformation on the two sides is larger than the weld center, so the residual stress is high. Fig. 7 shows longitudinal and transversal laser welding and low-cycle fatigue residual stress curves of the Ludwik-Voce Model. The residual stress is redistributed and relaxes after the low-cycle fatigue. When the cyclic loading superposes with the residual stress, the summed stress reaches or is higher than the yield strength, and plastic deformation occurs. In some cases, the summed stress does not reach the yield strength, but it causes local deformation of the welded structure. When the cyclic loading is removed, the welded structure undergoes secondary deformation, so the residual stress is redistributed and relaxes [START_REF] Benasciutti | Metal plasticity and fatigue at high temperature[END_REF] . Comparing the simulation and experimental results of low-cycle fatigue residual stress, as shown in Fig. 8, the trend is the same, but there are still some errors. It is because the fluid flow and phase transformations are neglected in the laser welding simulation, and the laser welding residual stress directly affects the low-cycle fatigue residual stress. Additionally, the initial residual stress present in the base metal is also ignored.

Conclusions

The following conclusions can be obtained from simulations and experiments for the residual stress after laser welding and low-cycle fatigue of the DP600 steel welded structure:

(1) The large temperature gradient and external loading can cause strain hardening. Constitutive models are used to simulate laser welding residual stress, and combined hardening models are used to simulate low-cycle fatigue residual stress.

(2) The laser welding residual stress reaches the maximum in the heat-affected zone. And the low-cycle fatigue residual stress on both sides of the welded structure is higher than the center along the longitudinal direction of the weld.

(3) Residual stress is redistributed and relaxes after low-cycle fatigue.

(4) Comparing the simulation and experimental results of low-cycle fatigue residual stress, the trend is the same.

Fig. 1 .

 1 Fig.1. Microstructure of DP600 steel.

Fig. 2 .

 2 Fig.2. Specimen geometry used for the finite element model.

Fig. 3 .

 3 Fig.3. The Conical heat source with Gaussian distribution.

Fig. 4 .

 4 Fig.4. Constitutive models fitting curves.

Fig. 5 .

 5 Fig.5. Longitudinal and transversal laser welding residual stress with three models (left) longitudinal residual stress; (right) transversal residual stress.

Fig. 6 .

 6 Fig.6. Longitudinal and transversal low-cycle fatigue residual stress curves with three models (left) longitudinal residual stress; (right) transversal residual stress.

Fig. 7 .

 7 Fig.7. Longitudinal and transversal laser welding and low-cycle fatigue residual stress (left) longitudinal residual stress; (right) transversal residual stress.

Fig. 8 .

 8 Fig.8. The comparison of simulation and experimental results of low-cycle fatigue residual stress.

Table 1 .

 1 Equations of constitutive models.

Table 2 .

 2 Parameters of constitutive models.

		σ0	K	K1	K2	n	n1	n2
	Ludwik Model	400 1244.24			0.53
	Voce Model	400 409.64			21.55

Acknowledgements

The work is financially supported by China Scholarship Council (CSC), INSA de Rennes and Laboratory of Digital Sciences of Nantes (LS2N).