Tensor-Train Isogeometric Analysis for Large-Scale Three-Dimensional Linear Elasticity
Abstract
This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.
Related
- Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics
- A Highly Scalable Quantized Tensor-Train FDTD Framework for the Simulation of Three-Dimensional Electromagnetic Scattering Problems
- Incremental Tensor-Train Compression from Streaming TT-Formatted Data with Applications to Reduced-Order Modeling of Linear Kinetic Equations
- A Local Stabilized Isogeometric Formulation with Discontinuous Pressure for Mesh-Conforming and Immersed Analysis of Nearly Incompressible Elasticity
- Dynamical Tensor Train Approximation for Wigner--Fokker--Planck Equation with Applications to Open Quantum Systems
BibTeX
@misc{tran2026tensortrain,
title = {{Tensor-Train Isogeometric Analysis for Large-Scale Three-Dimensional Linear Elasticity}},
author = {Quoc Thai Tran and Duc P. Truong and William Dai and Kim Ø. Rasmussen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23595},
eprint = {2608.23595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23595},
}
RIS
TY - UNPB
TI - Tensor-Train Isogeometric Analysis for Large-Scale Three-Dimensional Linear Elasticity
AU - Quoc Thai Tran
AU - Duc P. Truong
AU - William Dai
AU - Kim Ø. Rasmussen
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2608.23595
UR - https://arxiv.org/abs/2608.23595
AB - This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.
ER -