Revision as of 11:37, 20 July 2026

Abstract

This study introduces a novel coupled computational framework that seamlessly fuses the Isogeometric Boundary Element Method (IGABEM) with the Polynomial Chaos Expansion (PCE) to address transient heat conduction problems under parametric uncertainty. The proposed approach transcends traditional numerical paradigms by constructing a unified formulation that intertwines geometric precision, computational efficiency, and probabilistic rigor. At the heart of the framework lie three tightly integrated representations. First, Non-Uniform Rational B-Splines (NURBS) basis functions serve a dual purpose—they simultaneously embody geometric modeling and field discretization. This duality allows for the direct transformation of ComputerAided Design (CAD)-level geometric fidelity into the numerical domain, preserving intricate boundary characteristics while discretizing the governing boundary integral equations with remarkable exactness. Second, through the introduction of Bézier extraction, the computational overhead associated with NURBS basis evaluations is dramatically reduced. This operation reformulates complex NURBS structures into more manageable Bézier entities, unlocking substantial efficiency gains without compromising the inherent smoothness or precision of the geometric representation. Third, the framework incorporates Polynomial Chaos Expansion to systematically quantify and propagate uncertainties in material parameters. By coupling this stochastic representation with an advanced radial integration scheme, the method elegantly circumvents the need for conventional domain meshing while maintaining high accuracy in evaluating domain integrals. The result is a streamlined yet robust uncertainty quantification process embedded directly within the boundary element context. A series of numerical experiments corroborates the effectiveness of the proposed hybrid strategy. The findings reveal that the method not only preserves the celebrated advantages of IGABEM—such as geometric exactness and reduced-dimensional computation—but also extends its capability into the stochastic domain, enabling a comprehensive evaluation of how parameter uncertainties influence transient thermal responses. In essence, this integrated IGABEM–PCE framework bridges the gap between deterministic modeling and probabilistic analysis, yielding a unified, highfidelity numerical tool. Its capacity to deliver both geometric accuracy and uncertainty quantification positions it as an indispensable technique for complex engineering scenarios, particularly in thermal management, energy conversion, and high-precision design systems, where reliability under uncertainty is paramount.

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