An efficient radiation analysis approach through compressive model for laser driven inertial confinement fusion☆
Introduction
Controllable nuclear fusion is potential to solve the energy crisis in the future, and laser driven Inertial Confinement Fusion (ICF) is supposed to be one of the most promising ways [1]. To achieve this goal, the driven asymmetry on the centrally located capsule, which should be no more than 2% during the fusion process, needs to be evaluated. Due to the limited lasers beams, e.g. 192 laser beams for laser facility of National Ignition Facility (NIF), 2 laser entrance holes and a few discrete diagnose holes, the radiation asymmetry may be much larger than such prescribed threshold [2]. Therefore, we need to efficiently compute the driven flux reached the capsule to evaluate its driven asymmetry. In the ICF, the radiation flux on the fuel capsule is related to the laser-plasma interactions (LPI) and transport of x-rays from the cavity wall to the capsule, which involves the solving of complex kinetic and hydrodynamic equations implemented in the codes such as LASNEX [3]. In practice, simple mathematical models such as view-factor codes are usually employed to compute the radiation flux distributed on the capsule, especially for the preliminary design and optimization of thermonuclear target structure and shape [3], [4]. As described in [1], the radiation fluxed is usually obtained by solving a non-linear view-factor based equation model, which can be solved by utilizing Newton–Raphson [5], Jacobi iteration [6], Cholesky decomposition [7], or Preconditioned Conjugate Gradient method [8]. However, in order to improve the evaluation accuracy, the size of discrete element is usually very small, the number of elements or equations will increase significantly, which may lead to much time required to solve such large scale nonlinear equations. The computation time may be unacceptable for researchers. Therefore, a new efficient computation approach is essentially required to significantly improve the efficiency of radiation symmetry evaluation and optimization.
Compressed sensing (CS), proposed by Donoho and Candès [9], [10], is a new method to reconstruct signals from significantly fewer measurements than that required by traditional methods, which has attracted considerable attention and achieves successful applications such as Medical imaging (MI) [11], Analog-digital Conversion [12], Computational Biology [13], and Computer Graphics [14]. In the field of radiation symmetry evaluation of ICF, Compressive analysis approach has been applied to efficiently obtain the radiation flux distribution on the capsule [15]. However, the nonlinearity of radiation flux computation model is not taken into account. As a result, the evaluation accuracy of the radiation asymmetry is limited. Furthermore, only the sparse spherical harmonic basis is discussed to represent the radiation flux on the capsule. The non-linear Time Dependent Energy Balance Model (TDEBM) presented in [16] is often used to compute the radiation flux. Therefore, compressive sensing approach for non-linear equation solving approach, such as Iterative Hard Thresholding (IHT) [17], can be applied. However, the step size is often taken as a constant, which may lead to more times of iterations. Normalized Iterative Hard Thresholding (NIHT) [18] is further proposed, in which, fixed step length is replaced with a descending factor to accelerate the convergence rate. Nevertheless, the gradient based search direction in NIHT, may lead to very slow convergence when it approaches the minimum. Conjugate Gradient Iterative Hard Thresholding (CG-IHT) is proposed in [19], to replace the search direction with the conjugate gradient direction, which can accelerate the convergence. However, the search direction of CG-IHT may not be conjugate since the Jacobian matrix varies with each iteration, which still may lead to more times of iterations. In this paper, a Modified Conjugate Gradient Iterative Hard Thresholding algorithm (MCG-ITH) is proposed to largely reduce TDEBM and efficiently obtain the radiation flux. The core ideas include:
- (1)
The Legendre–Fourier basis, Zernike basis and spherical harmonics are employed to sparsely represent the radiation flux on the hohlraum and capsule, which enable only a few equations are required to recover the radiation flux with high accuracy, and significantly reduce the radiation computation model.
- (2)
Reduced nonlinear equation model with sparse coefficients is formulated to enable compressed sensing algorithms, such as CG-IHT, be used to efficiently obtain the radiation flux.
- (3)
Modified CG-IHT algorithm is then presented to enable the adjacent search direction be conjugate, which facilitate the convergence of iteration, rapidly obtain the solution, and then efficiently evaluate the radiation symmetry for ICF experiments design.
The remainder of this paper is organized as follows. Section 2 introduces the nonlinear TDEBM of ICF and the background of compressed sensing. In Section 3, the radiation flux is computed with the modified conjugate gradient iteration hard thresholding approach. In Section 4, the efficiency of the presented approach is validated with two actual experimental targets. This paper concludes with Section 5.
Section snippets
Radiation flux computation model in the ICF
As shown in the left of Fig. 1, eight laser beams are injected into the spherical cavity from two end entrance holes, and intersect with high-Z material (e.g. Au) on the wall of the cavity. The injected laser energy is converted into X-ray to irradiate the centrally located capsule. And the detailed energy relationship of the element of the spherical cavity or the capsule is shown in the right of Fig. 1. The input energy of the ith element includes the source energy from laser and the
Radiation flux computation based on MCG-IHT
In this section, the radiation flux computation based on MCG-IHT is introduced. First, the sparse representation of radiation flux is introduced, and then radiation symmetry analysis equations should be sampled. Last, the MCG-IHT is applied to obtain the sparse coefficients.
Numerical verification
In this section, both two experimental target on SGIII-YX laser facility build in China, are selected and run on the desktop computer (CPU: Intel Quad-core, 3.4 GHZ and memory: 8 G), and Matlab 2014a [31] is used to implement and compare the presented and usual approaches. The radiation faces of the capsule and cavity are faceted with three different mesh element sizes (corresponding to three kinds of mesh models S1, S2, S3).
As shown in Fig. 4, the overall model of SGIII-YX and a designed
Conclusion
The radiation flux distribution symmetry evaluation is very important in ICF experiments, which involve a time consumption process in solving the nonlinear energy equilibrium computation model. In order to accelerate such equation solving process, an efficient radiation computation approach is presented, in which (1) the sparsity of the radiation flux over the spherical harmonic domain is investigated for the capsule and the spherical cavity, only the coefficients less than order 25 are enough
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part National Natural Science Foundation of China #51975125, #51775116 and Creative Team Project of Foshan Nanhai Guangdong Technology University CNC equipment cooperative innovation Institute, China .
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The review of this paper was arranged by Prof. N.S. Scott.