{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T12:50:22Z","timestamp":1753879822967,"version":"3.41.2"},"reference-count":21,"publisher":"ASME International","issue":"1","content-domain":{"domain":["asmedigitalcollection.asme.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2013,3,1]]},"abstract":"<jats:p>Popular eigensolvers such as block-Lanczos require repeated inversion of an eigenmatrix. This is a bottleneck in large-scale modal problems with millions of degrees of freedom. On the other hand, the classic Rayleigh\u2013Ritz conjugate gradient method only requires a matrix-vector multiplication, and is therefore potentially scalable to such problems. However, as is well-known, the Rayleigh\u2013Ritz has serious numerical deficiencies, and has largely been abandoned by the finite-element community. In this paper, we address these deficiencies through subspace augmentation, and consider a subspace augmented Rayleigh\u2013Ritz conjugate gradient method (SaRCG). SaRCG is numerically stable and does not entail explicit inversion. As a specific application, we consider the modal analysis of geometrically complex structures discretized via nonconforming voxels. The resulting large-scale eigenproblems are then solved via SaRCG. The voxelization structure is also exploited to render the underlying matrix-vector multiplication assembly-free. The implementation of SaRCG on multicore central processing units (CPUs) and graphics-programmable units (GPUs) is discussed, followed by numerical experiments and case-studies.<\/jats:p>","DOI":"10.1115\/1.4023168","type":"journal-article","created":{"date-parts":[[2013,1,8]],"date-time":"2013-01-08T23:06:58Z","timestamp":1357686418000},"update-policy":"https:\/\/doi.org\/10.1115\/crossmarkpolicy-asme","source":"Crossref","is-referenced-by-count":4,"title":["Assembly-Free Large-Scale Modal Analysis on the Graphics-Programmable Unit"],"prefix":"10.1115","volume":"13","author":[{"given":"Praveen","family":"Yadav","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Krishnan","family":"Suresh","sequence":"additional","affiliation":[{"name":"e-mail:\u2002 Department of Mechanical Engineering, UW-Madison, Madison, WI 53706"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"33","published-online":{"date-parts":[[2013,1,7]]},"reference":[{"year":"2009","key":"2019100603052852900_B1","article-title":"A Survey of Software for Sparse Eigenvalue Problems"},{"issue":"2","key":"2019100603052852900_B2","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1002\/nme.1365","article-title":"A Comparison of Eigensolvers for Large-scale 3D Modal Analysis Using AMG-Preconditioned Iterative Methods","volume":"64","year":"2005","journal-title":"Int. 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