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Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods
Dissertation (MEng (Mechanical Engineering))--University of Pretoria, 2025.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2026
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| _version_ | 1869483775369412608 |
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| access_status_str | Open Access |
| author2 | Kok, Schalk |
| author_browse | Kok, Schalk |
| author_facet | Kok, Schalk |
| collection | Thesis |
| dc_rights_str_mv | © 2024 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Dissertation (MEng (Mechanical Engineering))--University of Pretoria, 2025. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/108374 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-07-01T04:04:21.508Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2026 |
| publishDateRange | 2026 |
| publishDateSort | 2026 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/108374 Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods Kok, Schalk u16006772@tuks.co.za Krüger, Stephan UCTD Sustainable Development Goals (SDGs) Gradient-enhanced surrogate modelling Radial basis functions Local Hessian Method Spatio-temporal transformations Anisotropy Dissertation (MEng (Mechanical Engineering))--University of Pretoria, 2025. Surrogate models are increasingly essential for high-fidelity engineering analysis, particularly when response surfaces exhibit anisotropy, strong variable coupling and temporal evolution. Classical radial basis function (RBF) surrogates, while flexible and meshfree, suffer from ill-conditioning and reduced accuracy under such conditions. This thesis develops a comprehensive framework for constructing accurate, stable and data-efficient surrogate models for anisotropic and spatio-temporal applications by integrating gradient-enhanced RBFs (GERBFs) with a curvature-informed coordinate transformation known as the Local Hessian Method (LHM). The research begins with a detailed assessment of Gaussian function value only RBF (FV-RBF) and GE-RBF formulations, demonstrating that gradient information substantially improves interpolation accuracy, especially under sparse sampling. However, the studies also reveal the limitations of classical isotropic kernels in anisotropic domains, motivating the development of transformation-based improvements. The LHM is introduced as a means of reconstructing a representative global Hessian from multiple locally estimated Hessians. The resulting eigenvalue–eigenvector decomposition provides a rotation–scaling transformation that restores isotropy and stabilises the surrogate model. Benchmark studies in two and four dimensions show that LHM-enhanced GE-RBFs match or exceed the performance of Gradient-Enhanced Kriging (GE-Kriging) and the Active Subspace Method (ASM), particularly for coupled anisotropic problems. A further contribution is the extension of the LHM to spatio-temporal domains, where dense temporal sampling creates non-uniform kernel distances and leads to ill-conditioning. The thesis proposes a combined strategy involving centre redistribution, reduced temporal sampling and an improved neighbour-selection scheme for curvature estimation. These developments enable the extraction of meaningful spatio-temporal curvature and yield substantial reductions in prediction error. The final part of the thesis applies the full GE-RBF with LHM framework to a nonlinear finite element analysis of boiler tubes with external erosion defects. Using a four-dimensional spatio-temporal dataset, the LHM-enhanced GE-RBF surrogate achieves an NRMSE of 0.86% and an R2 value of 0.999 on unseen data, outperforming both ASM-based GE-RBF and GE-Kriging at higher training densities. The surrogate is computationally inexpensive and suitable for real-time evaluation, enabling practical integration into inspection workflows for structural integrity assessment. Overall, the thesis establishes best-practice methodologies for constructing GE-RBF surrogates in anisotropic and spatio-temporal settings. The LHM is shown to be an effective and scalable approach for restoring isotropy, improving conditioning and enhancing predictive accuracy. The contributions provide a robust foundation for surrogate-assisted optimisation, structural assessment and future digital-twin applications. Mechanical and Aeronautical Engineering MEng (Mechanical Engineering) Unrestricted Faculty of Engineering, Built Environment and Information Technology None 2026-02-18T10:02:33Z 2026-02-18T10:02:33Z 2026-04-30 2025 Dissertation * A2026 http://hdl.handle.net/2263/108374 10.25403/UPresearchdata.31354579 en © 2024 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | UCTD Sustainable Development Goals (SDGs) Gradient-enhanced surrogate modelling Radial basis functions Local Hessian Method Spatio-temporal transformations Anisotropy Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| title | Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| title_full | Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| title_fullStr | Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| title_full_unstemmed | Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| title_short | Efficient construction of spatio-temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| title_sort | efficient construction of spatio temporal surrogates using gradient enhanced radial basis functions and coordinate transformation methods |
| topic | UCTD Sustainable Development Goals (SDGs) Gradient-enhanced surrogate modelling Radial basis functions Local Hessian Method Spatio-temporal transformations Anisotropy |
| url | http://hdl.handle.net/2263/108374 |