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Being a data-driven method, Cubature requires a set of representative deformed poses that cover deformations likely to occur in the simulation. This requirement is unlikely for high-resolution deformation with rich local details. Therefore, simulation performance and convergence degenerate when Cubature extrapolates. This paper proposes a training-free subspace integration algorithm based on classic Gaussian quadrature (GQ). We leverage the fact that the subproblem's subspace bases can be well-approximated by a low-degree multivariable polynomial, which suggests GQ an excellent candidate for Cubature substitute. To this end, we introduce a novel algorithm that adaptively generates the integration region for each subproblem. As a result, GQ integration can be analytically retrieved without cumbersome data generation and training. We also show how to handle frictional contact by modifying the pre-computed perturbation subspace. The resulting JGS2-GQ framework is more versatile than the vanilla JGS2 method. It is more stable for large and novel deformations, and is free of data generation and expensive training, while maintaining a near second-order convergence that is comparable to Newton. Performance-wise, JGS2-GQ is as efficient as JGS2, which is three orders faster than classic CPU methods and up to two orders faster than classic GPU algorithms. When novel deformation occurs, JGS2-GQ outperforms JGS2 over 50%.<\/jats:p>","DOI":"10.1145\/3811274","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T07:05:51Z","timestamp":1783062351000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["JGS2-GQ: Training-free 2nd Jacobi with Gaussian Quadrature"],"prefix":"10.1145","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7393-5486","authenticated-orcid":false,"given":"Dewen","family":"Guo","sequence":"first","affiliation":[{"name":"University of Utah, Salt Lake City, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0067-0242","authenticated-orcid":false,"given":"Zixuan","family":"Lu","sequence":"additional","affiliation":[{"name":"University of Utah, Salt Lake City, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-7198-0443","authenticated-orcid":false,"given":"Zhiyong","family":"He","sequence":"additional","affiliation":[{"name":"University of Utah, Salt Lake City, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-4980-2180","authenticated-orcid":false,"given":"Yuqi","family":"Meng","sequence":"additional","affiliation":[{"name":"University of Utah, Salt Lake City, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1439-1455","authenticated-orcid":false,"given":"Bohan","family":"Wang","sequence":"additional","affiliation":[{"name":"National University of Singapore, Singapore, Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-7626-7580","authenticated-orcid":false,"given":"Lei","family":"Lan","sequence":"additional","affiliation":[{"name":"State Key Lab of CAD and CG, Zhejiang University, Zhejiang, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3756-3539","authenticated-orcid":false,"given":"Weiwei","family":"Xu","sequence":"additional","affiliation":[{"name":"State Key Lab of CAD and CG, Zhejiang University, Zhejiang, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3506-0583","authenticated-orcid":false,"given":"Chenfanfu","family":"Jiang","sequence":"additional","affiliation":[{"name":"UCLA, Los Angeles, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7645-5931","authenticated-orcid":false,"given":"Yin","family":"Yang","sequence":"additional","affiliation":[{"name":"University of Utah, Salt Lake City, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2026,7,3]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"Tomas Akenine-M\u00f6ller. 2005. 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