Sub-matrix Method Research Articles

A modified semi-weight function method for SIF calculation of arbitrary inclined interface crack with both single complex-root and double real-root singularities

An arbitrary inclined crack terminated at the interface usually has singularity of single complex-root or double real-root at crack tip. Currently, most methods of the stress intensity factor (SIF) calculation (e.g., Mellin transformation method, enrichment strategy method) only consider the single complex-root singularity, with difficulty in solving auxiliary functions. Although the scaled boundary method takes the double real-root singularity into account, it needs to set special finite elements at crack tip. In this study, based on the sub-matrix method and the reciprocal theorem of work, a modified semi-weight function method is proposed to calculate SIF of the arbitrary interface crack with consideration of both single complex-root and double real-root singularity, and also analyze the effect of bimaterial parameters (Gi), crack inclination angle (w) and relative length (a/L) on the SIFs. This modified method has the advantages of uniform auxiliary function (independent of crack size and external load) and simple finite element calculation (without special singularity element). Moreover, its validity is verified by good agreement of our calculation results with the available literature results and finite element results. The research results can provide a theoretical basis for anti-crack design of multi-layer caverns of the compressed air energy storage (CAES).

Breaking the exascale barrier for the electronic structure problem in ab-initio molecular dynamics

The non-orthogonal local submatrix method applied to electronic structure–based molecular dynamics simulations is shown to exceed 1.1 EFLOP/s in FP16/FP32-mixed floating-point arithmetic when using 4400 NVIDIA A100 GPUs of the Perlmutter system. This is enabled by a modification of the original method that pushes the sustained fraction of the peak performance to about 80%. Example calculations are performed for SARS-CoV-2 spike proteins with up to 83 million atoms.

Towards electronic structure-based ab-initio molecular dynamics simulations with hundreds of millions of atoms

We push the boundaries of electronic structure-based ab-initio molecular dynamics (AIMD) beyond 100 million atoms. This scale is otherwise barely reachable with classical force-field methods or novel neural network and machine learning potentials. We achieve this breakthrough by combining innovations in linear-scaling AIMD, efficient and approximate sparse linear algebra, low and mixed-precision floating-point computation on GPUs, and a compensation scheme for the errors introduced by numerical approximations. The core of our work is the non-orthogonalized local submatrix method (NOLSM), which scales very favorably to massively parallel computing systems and translates large sparse matrix operations into highly parallel, dense matrix operations that are ideally suited to hardware accelerators. We demonstrate that the NOLSM method, which is at the center point of each AIMD step, is able to achieve a sustained performance of 324 PFLOP/s in mixed FP16/FP32 precision corresponding to an efficiency of 67.7% when running on 1536 NVIDIA A100 GPUs.

A modified semi-weight function method for stress intensity factor calculation of a vertical crack terminating at the interface

The interfacial crack stress intensity factor (SIF) calculation plays an important role in the safety evaluation and anti-crack design for multilayer structures. The current analytical or numerical methods have difficulty of complicated auxiliary functions or special crack-tip singular elements. In this study, a modified semi-weight function method was proposed based on the submatrix method to calculate the SIFs of the crack vertical to and terminating at the interface of bimaterial (easily appears in compressed air energy storage) and verified by two calculation examples compared with the available literature results. The effects of bimaterial parameters (Ei, υi), relative crack length (a/w2) and outer integration path (r2) on the SIFs were also analyzed. Research results show that the SIFs (K1 and K2) are related to only the ratio of E2/E1 independent of the single E1 or E2, and they are increased with increase of a/w2. The reasonable range of r2 is rs < r2 < a/2 (rs is minimum finite element size). The modified semi-weight function method has advantages of unified auxiliary function for different crack sizes and external loads and simpler calculation process without any crack-tip singular elements over current methods.

A Random Algorithm for Low-Rank Decomposition of Large-Scale Matrices With Missing Entries.

A random submatrix method (RSM) is proposed to calculate the low-rank decomposition U(m×r)V(n×r)(T) (r < m, n) of the matrix Y∈R(m×n) (assuming m > n generally) with known entry percentage 0 < ρ ≤ 1. RSM is very fast as only O(mr(2)ρ(r)) or O(n(3)ρ(3r)) floating-point operations (flops) are required, compared favorably with O(mnr+r(2)(m+n)) flops required by the state-of-the-art algorithms. Meanwhile, RSM has the advantage of a small memory requirement as only max(n(2),mr+nr) real values need to be saved. With the assumption that known entries are uniformly distributed in Y, submatrices formed by known entries are randomly selected from Y with statistical size k×nρ(k) or mρ(l)×l , where k or l takes r+1 usually. We propose and prove a theorem, under random noises the probability that the subspace associated with a smaller singular value will turn into the space associated to anyone of the r largest singular values is smaller. Based on the theorem, the nρ(k)-k null vectors or the l-r right singular vectors associated with the minor singular values are calculated for each submatrix. The vectors ought to be the null vectors of the submatrix formed by the chosen nρ(k) or l columns of the ground truth of V(T). If enough submatrices are randomly chosen, V and U can be estimated accordingly. The experimental results on random synthetic matrices with sizes such as 13 1072 ×10(24) and on real data sets such as dinosaur indicate that RSM is 4.30 ∼ 197.95 times faster than the state-of-the-art algorithms. It, meanwhile, has considerable high precision achieving or approximating to the best.

Yet another storage technique for lr parsing tables

AbstractA submatrix storage scheme for sparse matrices is presented which has been found useful for storing LR parsing tables. It is not the most compact representation but it is easy to use and of order (1) in speed of access. The method is compared with the two most popular methods of storing LR parsing tables which use ‘linear lists’ and ‘row displacement’. Results of using the submatrix method to store the parsing tables for the Pascal language are included.