Large-scale finite element analysis (FEA) with millions of degrees of freedom (DOF) is becoming commonplace in solid mechanics. The primary computational bottleneck in such problems is the solution of large linear systems of equations. In this paper, we propose an assembly-free version of the deflated conjugate gradient (DCG) for solving such equations, where neither the stiffness matrix nor the deflation matrix is assembled. While assembly-free FEA is a well-known concept, the novelty pursued in this paper is the use of assembly-free deflation. The resulting implementation is particularly well suited for large-scale problems and can be easily ported to multicore central processing unit (CPU) and graphics-programmable unit (GPU) architectures. For demonstration, we show that one can solve a 50 × 106 degree of freedom system on a single GPU card, equipped with 3 GB of memory. The second contribution is an extension of the “rigid-body agglomeration” concept used in DCG to a “curvature-sensitive agglomeration.” The latter exploits classic plate and beam theories for efficient deflation of highly ill-conditioned problems arising from thin structures.

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Linear buckling analysis entails the solution of a generalized eigenvalue problem. Popular methods for solving such problems tend to be memory-hungry, and therefore slow for large degrees of freedom. The main contribution of this paper is a low-memory assembly-free linear buckling analysis method. In particular, we employ the classic inverse iteration, in conjunction with an assembly-free deflated linear solver. The resulting implementation is simple, fast and particularly well suited for parallelization. The proposed method is used here to solve large scale 3D topology optimization problems subject to buckling constraints, where buckling problems must be solved repeatedly.

The coherent density fluctuation method (CDFM) is used to estimate the isospin-dependent properties of finite nuclei such as symmetry energy, surface symmetry energy, and volume symmetry energy from its corresponding component in infinite nuclear matter. The relativistic mean-field (RMF) formalism with non-linear NL3 and Relativistic-Hartree-Bogoliubov with density-dependent DD-ME2 interaction parameters are employed in the present analysis. The weight function $\vert \mathcal{F}(x) \vert^{2} $ is estimated using the total density of each nucleus, which in turn is used along with the nuclear matter quantities to obtain the effective symmetry energy and its components in the finite nuclei. We calculate the ground state bulk properties such as nuclear binding energy, quadrupole deformation, two-neutron separation energy, the differential variation of two-neutron separation energy, and root-mean-square charge radius for the Sc- and Ti- isotopic chains based on non-linear NL3 and density-dependent DD-ME2 parameter sets. Further, the ground state density distributions are used within the CDFM to obtain the effective surface properties such as symmetry energy and its components, namely volume and surface symmetry energy, for both the parameter sets. The calculated quantities are used to understand the isospin dependent structural properties of finite nuclei near and beyond the drip line that will further our horizon of finding newer magicity along the isotopic chains. A shape transition is observed from prolate-spherical-prolate near $N$ = 40 for Sc-, and spherical to prolate near the drip-line region of Ti- Isotopes. Notable signatures of shell and/or sub-shell closures have been found for the magic neutron numbers at $N$ = 20 and 28 for both the isotopic chain using the nuclear bulk and isospin quantities. In addition to these, a few signatures of shell/sub-shell closure are observed near the drip-line region, at $N$ = 34 and 50 by following the isospin dependent observables, namely symmetry energy and its components.