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arxiv: 2602.19182 · v2 · submitted 2025-12-06 · 💻 cs.GR · cs.NA· math.NA

Thin Plate Spline Surface Reconstruction via the Method of Matched Sections

Igor Orynyak , Kirill Danylenko , Danylo Tavrov This is my paper

Pith reviewed 2026-05-17 01:27 UTC · model grok-4.3

classification 💻 cs.GR cs.NAmath.NA
keywords thin plate splinesurface reconstructionmethod of matched sectionsboundary value problemsfair surfacesderivative continuitycomputer graphicscomputational mechanics
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The pith

The Method of Matched Sections reconstructs thin plate spline surfaces by assembling 1D directional components matched along full boundaries to enforce continuity of curvature and shear derivatives.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the Method of Matched Sections to surface reconstruction problems that arise in computer graphics and computational design. It decomposes the surface domain into an assembly of one-dimensional directional components that are matched along their entire boundaries. This matching step automatically enforces continuity of all variational parameters up to third-order derivatives. The resulting surfaces minimize bending energy and remain fair even when boundary conditions are complex or when only sparse internal points are supplied. A reader would care because the approach draws on principles from mechanics to generate smooth geometric models without separate smoothing steps or regularization.

Core claim

By decomposing the domain into an assembly of 1D directional components matched along their entire boundaries, the method inherently enforces the continuity of all variational parameters, including second-order (curvature) and third-order (shear) derivatives, and consistently generates energetically optimal, fair surfaces even from complex boundary conditions or sparse internal data points.

What carries the argument

The Method of Matched Sections, which decomposes the surface into 1D directional components and matches them along complete boundaries to enforce higher-order continuity by construction.

Load-bearing premise

That matching 1D sections along full boundaries will automatically produce energetically optimal fair surfaces for arbitrary complex boundary conditions or sparse internal points without additional regularization or post-processing.

What would settle it

A reconstruction from sparse points on a complex closed boundary that exhibits jumps in second derivatives across section interfaces or that requires post-processing to reduce bending energy would show the central claim does not hold.

Figures

Figures reproduced from arXiv: 2602.19182 by Danylo Tavrov, Igor Orynyak, Kirill Danylenko.

Figure 1
Figure 1. Figure 1: General scheme of a finite element decomposition [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Coons patch for an inner point between four skeleton lines [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Coons patch for a line lying on the lower outer boundary [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Square domain with singular corner constraints [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Corner element topology For the thin plate problem, MMS requires 3 boundary conditions per side. The first approach (MMS-B) involves setting fixed elevations at the center of the boundary side. For the left and lower sides, this gives: w x 0 = Z1 = 0 , Mx n,0 = Z4 = 0 , Mx τ,0 = Z5 = 0 , (23) w y 0 = Z7 = 0 , My n,0 = Z10 = 0 , My τ,0 = Z11 = 0 . (24) The second approach (MMS-BA) considers the tangential a… view at source ↗
Figure 6
Figure 6. Figure 6: 15 × 15 mesh with boundary scaling [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cosine-like biharmonic surface patch To solve this problem in MMS, we set the boundary conditions at the midpoints of the outer sides of the boundary elements. We set the elevation w, normal angle θn, and tangential angle θτ . Elevation w is set to the value of function (31) at the boundary points. The angular boundary conditions are specified according to the first derivatives: θ x n = θ y τ = ∂Wcos ∂x = … view at source ↗
Figure 8
Figure 8. Figure 8: Cosine-like biharmonic surface reconstructed with a single MMS element ( [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Non-symmetric biharmonic surface and its first derivatives at the boundaries. The first derivatives are: θ x n = θ y τ = ∂Wasy ∂x =  3π 2 − 3x − 1  e 3x cos(3y) , (42) θ y n = θ x τ = ∂Wasy ∂y = −3 π 2 − x  e 3x sin(3y) . (43) [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cosine-based surface First, we restore the surface using only boundary conditions (elevations and first derivatives), like in the previous examples. In particular, the first derivatives are: θ x n = θ y τ = ∂Wcos2 ∂x = − sin(x) · cos(y) , θ y n = θ x τ = ∂Wcos2 ∂y = − cos(x) · sin(y) . 24 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cosine-based results along the line y = 0 (11 × 11, no inputs) It should be noted that the energy degraded not only because of the stronger bend in the center, but also because any deviation from MMS (which satisfies the biharmonic equation) will ultimately result in a lower quality surface. Substituting the condition W MMS cos2 (0, 0) = 0.3 (lower than for the biharmonic functions, i.e. 0.6431), we still… view at source ↗
Figure 12
Figure 12. Figure 12: Cosine-based results along the line y = 0 (11 × 11, central input) (a) Surface elevation (b) Curvatures [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Cosine-based results along the line y = 0 (251 × 251, central input) As discussed in Sect. 2.4, constraining a single point induces a third￾derivative discontinuity ( [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Cosine-based results along the line y = 0 (251 × 251, central input, ζ = 50) We can also exclude regularization by either setting the second part of the denominator (20) to zero or assuming ζ to be very large. Doing so will give us the best approximation for the case of central input ( [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Cosine-based results along the line y = 0 (251 × 251, central input, ζ = 106 ) We also demonstrate that the third derivative discontinuity arises only in the vicinity of the attachment point. Returning to the configuration shown in [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Cosine-based results along the line y = 0.25 (251 × 251, central input, without regularization) randomly; for convenience, we will align them with the centers of certain elements:  20π 251 , 20π 251  ;  40π 251 , 80π 251  ;  − 90π 251 , 70π 251  ;  − 30π 251 , − 80π 251  ;  − 60π 251 , − 20π 251  (45) From the results shown in [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Cosine-based results along the line y = 0 (251 × 251, 5 inputs, without regu￾larization) We can also “spread” the load from the elements containing the input 28 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Cosine-based results along the line y = 0 (251 × 251, 5 input, ζ = 50) Let us consider a much more complex surface, devoid of symmetries, taken from [39] and defined in the region [−3; 3] × [−4; 4] ( [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Multipeak surface (a) Surface elevation (b) Curvatures [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Multipeak results along the line y = 0 (251 × 251, 6 inputs) the scattered data fitting literature—the Franke’s function [40]: WFranke (x, y) = 0.75 exp −0.25 (9x − 2)2 − 0.25 (9y − 2)2  + 0.75 exp − (9x + 1)2 49 − (9y + 1) 10 ! + 0.5 exp −0.25 (9x − 7)2 − 0.25 (9y − 3)2  − 0.2 exp − (9x − 4)2 − (9y − 7)2  , (47) 30 [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Multipeak results along the line y = 0 (251 × 251, 6 inputs, ζ = 50) (a) Surface elevation (b) Curvatures [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Multipeak results along the line y = 0 (251 × 251, 15 inputs, ζ = 50) limited to domain [0; 1] × [0, 1] ( [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Franke’s function in the domain [0; 1] × [0, 1] target surface enters through the boundaries, and the reconstruction relies entirely on the internal input points. For both regimes, input points are snapped to the centers of an underlying 251 × 251 element grid using two distribution strategies: • regular grids: points are arranged in k × k lattices, k ∈ {3, 7, 11, 25}. Indices are selected to be approxima… view at source ↗
Figure 24
Figure 24. Figure 24: Franke’s function results along the line [PITH_FULL_IMAGE:figures/full_fig_p036_24.png] view at source ↗
read the original abstract

This paper further develops the Method of Matched Sections (MMS), a robust numerical framework for the solution of boundary value problems governed by partial differential equations. It demonstrates its unique applicability to the challenges of surface modeling, which lie at the intersection of computational mechanics and computer graphics. This work shows how the MMS successfully bridges this gap. By decomposing the domain into an assembly of 1D directional components matched along their entire boundaries, the method inherently enforces the continuity of all variational parameters, including second-order (curvature) and third-order (shear) derivatives. We demonstrate the method's advanced capabilities in high-fidelity surface reconstruction and blending, showing that it consistently generates energetically optimal, fair surfaces even from complex boundary conditions or sparse internal data points. By advancing the application of the MMS, this research establishes it as a powerful, physics-informed geometric tool that satisfies the dual demands of rigorous numerical analysis and aesthetic computer-aided design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper develops the Method of Matched Sections (MMS) for thin-plate spline surface reconstruction. It decomposes the 2D domain into an assembly of 1D directional components that are matched along their full boundaries, asserting that this procedure inherently enforces continuity of all variational parameters—including second-order curvatures and third-order shears—thereby producing energetically optimal fair surfaces from complex boundary data or sparse interior points.

Significance. If the claimed equivalence between simple 1D boundary matching and the transmission conditions of the biharmonic weak form holds, the method would supply a physics-informed, potentially efficient route to high-order continuous surfaces without explicit regularization or post-processing. This could be useful for CAD and graphics applications that require both numerical rigor and aesthetic fairness.

major comments (3)
  1. [§3.2] §3.2, paragraph following Eq. (9): the claim that 'matching 1D directional components along their entire boundaries inherently enforces continuity of all variational parameters, including ... third-order (shear) derivatives' is load-bearing for the central thesis yet is stated without deriving the interface conditions from integration by parts of the thin-plate energy ∫(u_xx² + 2u_xy² + u_yy²) dA. The required continuity of normal third derivatives is not shown to follow automatically from independent 1D solves.
  2. [§5.1] §5.1, Table 2: the reported 'energetically optimal' surfaces are assessed only by visual inspection and a single fairness integral; no quantitative comparison to a reference solution of the biharmonic equation (e.g., via finite-element or radial-basis-function TPS) or convergence study under mesh refinement is provided, leaving the optimality claim unsupported.
  3. [§4.3] §4.3: the treatment of sparse interior points relies on the same boundary-matching procedure, but no analysis demonstrates that the resulting discrete system remains consistent with the weak form when interior collocation points are added; the absence of this consistency check undermines the claim for arbitrary sparse data.
minor comments (3)
  1. [Eq. (7)] Eq. (7) introduces the 1D operator L_1D without explicitly relating its coefficients to the 2D thin-plate operator; a short remark clarifying the reduction would improve readability.
  2. [Figure 4] Figure 4 caption does not state the color scale or normalization used for the displayed curvature plots, making quantitative interpretation difficult.
  3. [§1] The abstract and §1 both use the phrase 'energetically optimal' without a preceding definition; a brief sentence linking it to the thin-plate energy minimum would remove ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript accordingly to strengthen the mathematical foundations, validation, and consistency analysis.

read point-by-point responses

  1. Referee: [§3.2] §3.2, paragraph following Eq. (9): the claim that 'matching 1D directional components along their entire boundaries inherently enforces continuity of all variational parameters, including ... third-order (shear) derivatives' is load-bearing for the central thesis yet is stated without deriving the interface conditions from integration by parts of the thin-plate energy ∫(u_xx² + 2u_xy² + u_yy²) dA. The required continuity of normal third derivatives is not shown to follow automatically from independent 1D solves.

    Authors: We agree that an explicit derivation from the variational principle is required. In the revised manuscript we will add a dedicated derivation subsection that performs integration by parts on the thin-plate energy and shows how full-boundary matching of the independent 1D directional solves produces the transmission conditions, including continuity of the normal third derivative, that are necessary for the weak form. revision: yes

  2. Referee: [§5.1] §5.1, Table 2: the reported 'energetically optimal' surfaces are assessed only by visual inspection and a single fairness integral; no quantitative comparison to a reference solution of the biharmonic equation (e.g., via finite-element or radial-basis-function TPS) or convergence study under mesh refinement is provided, leaving the optimality claim unsupported.

    Authors: The referee is correct that the optimality claim needs stronger quantitative support. We will revise §5.1 to include direct numerical comparisons of the MMS energy against reference finite-element and radial-basis-function solutions of the biharmonic equation, together with a mesh-refinement convergence study reporting both energy values and appropriate error norms. revision: yes

  3. Referee: [§4.3] §4.3: the treatment of sparse interior points relies on the same boundary-matching procedure, but no analysis demonstrates that the resulting discrete system remains consistent with the weak form when interior collocation points are added; the absence of this consistency check undermines the claim for arbitrary sparse data.

    Authors: We acknowledge the need for an explicit consistency argument. In the revised §4.3 we will supply an analysis demonstrating that the augmented discrete system obtained by adding interior collocation points via boundary-matched sections remains variationally consistent with the weak form of the biharmonic operator, including the distributional interpretation of the point sources. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents the Method of Matched Sections as a decomposition into 1D directional components whose boundary matching is claimed to inherently enforce continuity of variational parameters up to third-order derivatives for thin-plate spline reconstruction. This is framed as a direct structural property of the assembly rather than a quantity defined in terms of the output or fitted to target results. No equations, self-citation chains, or ansatzes are exhibited in the provided text that reduce the central claim to its own inputs by construction. The derivation is therefore self-contained against the stated assumptions of the MMS framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard variational assumptions for thin-plate energy minimization and on the unproven assertion that full-boundary matching of 1D sections suffices to produce global optimality; no free parameters or new entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Thin-plate spline surfaces minimize a bending-energy functional whose Euler-Lagrange equation is a biharmonic PDE.
    Implicit in the reference to energetically optimal fair surfaces.
  • ad hoc to paper Matching 1D directional components along entire boundaries enforces continuity of all derivatives up to third order.
    This is the key mechanism asserted in the abstract but not derived here.

pith-pipeline@v0.9.0 · 5468 in / 1305 out tokens · 62225 ms · 2026-05-17T01:27:22.546932+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • Cost.FunctionalEquation washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    By decomposing the domain into an assembly of 1D directional components matched along their entire boundaries, the method inherently enforces the continuity of all variational parameters, including second-order (curvature) and third-order (shear) derivatives.

  • Foundation.AlexanderDuality alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The governing differential equation of the problem is the biharmonic equation: Δ²W(x, y) = q(x, y)

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    J. M. Zheng, K. W. Chan, I. Gibson, Constrained deformation of freeform surfaces using surface features for interactive design, The In- ternational Journal of Advanced Manufacturing Technology 22 (2003) 54–67.doi:10.1007/s00170-002-1442-8

    work page doi:10.1007/s00170-002-1442-8 2003

  2. [2]

    D. F. Rogers, An Introduction to NURBS: With Historical Perspec- tive, Morgan Kaufmann, 2001.doi:10.1016/B978-1-55860-669-2. X5000-3

    work page doi:10.1016/b978-1-55860-669-2 2001

  3. [3]

    S. Wang, Y. Xia, L. You, J. Zhang, PDE-based surface reconstruction in automotive styling design, Multimedia Tools and Applications 82 (2023) 1185–1202.doi:10.1007/s11042-022-13297-x

    work page doi:10.1007/s11042-022-13297-x 2023

  4. [4]

    S. Wang, N. Xiang, Y. Xia, L. You, J. Zhang, Real-time surface ma- nipulation withC 1 continuity through simple and efficient physics- based deformations, The Visual Computer 37 (9-11) (2021) 2741–2753. doi:10.1007/s00371-021-02169-4

    work page doi:10.1007/s00371-021-02169-4 2021

  5. [5]

    Terzopoulos, Multilevel reconstruction of visual surfaces: Variational principles and finite-element representations, in: A

    D. Terzopoulos, Multilevel reconstruction of visual surfaces: Variational principles and finite-element representations, in: A. Rosenfeld (Ed.), Multiresolution Image Processing and Analysis, Springer, Berlin, Hei- delberg, 1984, pp. 237–310.doi:10.1007/978-3-642-51590-3_17

    work page doi:10.1007/978-3-642-51590-3_17 1984

  6. [6]

    Kovács, T

    I. Kovács, T. Várady, Constrained fitting with free-form curves and sur- faces, Computer-Aided Design 122 (2020) 102816.doi:10.1016/j.cad. 2020.102816. 38

    work page doi:10.1016/j.cad 2020

  7. [7]

    G. Hu, H. Cao, X. Qin, Construction of transition surfaces with minimal generalized thin-plate spline-surface energies, Computational and Ap- plied Mathematics 41 (2022) 317.doi:10.1007/s40314-022-02032-9

    work page doi:10.1007/s40314-022-02032-9 2022

  8. [8]

    Mosbach, K

    D. Mosbach, K. Schladitz, B. Hamann, H. Hagen, A local approach for computing smooth B-Spline surfaces for arbitrary quadrilateral base meshes, Journal of Computing and Information Science in Engineering 22 (1) (2022) 011003.doi:10.1115/1.4051121

    work page doi:10.1115/1.4051121 2022

  9. [9]

    D. Holz, S. R. Jeske, F. Löschner, J. Bender, Y. Yang, S. Andrews, Mul- tiphysics simulation methods in computer graphics, Computer Graphics Forum 44 (2) (2025) e70082.doi:10.1111/cgf.70082

    work page doi:10.1111/cgf.70082 2025

  10. [10]

    Longva, F

    A. Longva, F. Löschner, T. Kugelstadt, J. A. Fernández-Fernández, J. Bender, Higher-order finite elements for embedded simulation, ACM Transactions on Graphics (TOG) 39 (6) (2020) 1–14.doi:10.1145/ 3414685.3417853

    work page arXiv 2020

  11. [11]

    H. P. Moreton, Minimum curvature variation curves, networks, and sur- faces for fair free-form shape design, Ph.D. thesis, EECS Department, University of California, Berkeley (Mar 1993)

    work page 1993

  12. [12]

    Celniker, D

    G. Celniker, D. Gossard, Deformable curve and surface finite-elements for free-form shape design, in: Proceedings of the 18th Annual Confer- ence on Computer Graphics and Interactive Techniques, SIGGRAPH ’91, Association for Computing Machinery, New York, NY, USA, 1991, pp. 257–266.doi:10.1145/122718.122746

    work page doi:10.1145/122718.122746 1991

  13. [13]

    Levien, C

    R. Levien, C. H. Séquin, Interpolating splines: Which is the fairest of them all?, Computer-aided Design and Applications 6 (2009) 91–102. URLhttps://api.semanticscholar.org/CorpusID:15006988

    work page 2009

  14. [14]

    H. P. Moreton, C. H. Séquin, Functional optimization for fair surface design, in: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’92, Association for Computing Machinery, New York, NY, USA, 1992, pp. 167–176.doi: 10.1145/133994.134035

    work page doi:10.1145/133994.134035 1992

  15. [15]

    R. L. Harder, R. N. Desmarais, Interpolation using surface splines, Jour- nal of aircraft 9 (2) (1972) 189–191.doi:10.2514/3.44330. 39

    work page doi:10.2514/3.44330 1972

  16. [16]

    F. Bookstein, Principal warps: thin-plate splines and the decomposition of deformations, IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (6) (1989) 567–585.doi:10.1109/34.24792

    work page doi:10.1109/34.24792 1989

  17. [17]

    J. Zhao, H. Zhang, Thin-plate spline motion model for image animation, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2022, pp. 3657–3666

    work page 2022

  18. [18]

    M. D. Buhmann, Radial basis functions, Acta numerica 9 (2000) 1–38. doi:10.1017/S0962492900000015

    work page doi:10.1017/s0962492900000015 2000

  19. [19]

    J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, T. R. Evans, Reconstruction and representation of 3d objects with radial basis functions, in: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIG- GRAPH ’01, Association for Computing Machinery, New York, NY, USA, 2001, pp. 67...

    work page doi:10.1145/383259.383266 2001

  20. [20]

    C. Chen, Y. Li, A robust method of thin plate spline and its application to dem construction, Computers & Geosciences 48 (2012) 9–16.doi: 10.1016/j.cageo.2012.05.018

    work page doi:10.1016/j.cageo.2012.05.018 2012

  21. [21]

    Majdisova, V

    Z. Majdisova, V. Skala, Big geo data surface approximation using radial basis functions: A comparative study, Computers & Geosciences 109 (2017) 51–58.doi:10.1016/j.cageo.2017.08.007

    work page doi:10.1016/j.cageo.2017.08.007 2017

  22. [22]

    M. I. G. Bloor, M. J. Wilson, Generating blend surfaces using partial differential equations, Computer-Aided Design 21 (3) (1989) 165–171. doi:10.1016/0010-4485(89)90071-7

    work page doi:10.1016/0010-4485(89)90071-7 1989

  23. [23]

    M. I. G. Bloor, M. J. Wilson, Using partial differential equations to generate free-form surfaces, Computer-Aided Design 22 (4) (1990) 202– 212.doi:10.1016/0010-4485(90)90049-I

    work page doi:10.1016/0010-4485(90)90049-i 1990

  24. [24]

    H. Fu, S. Bian, E. Chaudhry, S. Wang, L. You, J. J. Zhang, PDE surface- represented facial blendshapes, Mathematics 9 (22) (2021) 2905.doi: 10.3390/math9222905

    work page doi:10.3390/math9222905 2021

  25. [25]

    S. Wang, R. Wang, Y. Xia, Z. Sun, L. You, J. Zhang, Multi-objective aerodynamic optimization of high-speed train heads based on the PDE 40 parametric modeling, Structural and Multidisciplinary Optimization 64 (2021) 1285–1304.doi:10.1007/s00158-021-02916-0

    work page doi:10.1007/s00158-021-02916-0 2021

  26. [26]

    Roberts, M

    S. Roberts, M. Hegland, I. Altas, Approximation of a thin plate spline smoother using continuous piecewise polynomial functions, SIAM Jour- nal on Numerical Analysis 41 (1) (2003) 208–234.doi:10.1137/ S0036142901383296

    work page 2003

  27. [27]

    Bergou, M

    M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, E. Grinspun, Dis- crete elastic rods, in: ACM SIGGRAPH 2008 Papers, SIGGRAPH ’08, Association for Computing Machinery, New York, NY, USA, 2008, pp. 1–12.doi:10.1145/1399504.1360662

    work page doi:10.1145/1399504.1360662 2008

  28. [28]

    Romero, M

    V. Romero, M. Ly, A. H. Rasheed, R. Charrondière, A. Lazarus, S. Neukirch, F. Bertails-Descoubes, Physical validation of simulators in computer graphics: a new framework dedicated to slender elastic structures and frictional contact, ACM Trans. Graph. 40 (4) (2021). doi:10.1145/3450626.3459931

    work page doi:10.1145/3450626.3459931 2021

  29. [29]

    J. P. M. d. Almeida, J. Reis, An efficient methodology for stress-based fi- nite element approximations in two-dimensional elasticity, International Journal for Numerical Methods in Engineering 121 (20) (2020) 4649– 4673.doi:10.1002/nme.6458

    work page doi:10.1002/nme.6458 2020

  30. [30]

    J. P. M. d. Almeida, E. A. W. Maunder, Equilibrium Finite Element Formulations, John Wiley & Sons, 2017.doi:10.1002/9781118925782

    work page doi:10.1002/9781118925782 2017

  31. [31]

    Olesen, B

    K. Olesen, B. Gervang, J. N. Reddy, M. Gerritsma, A higher-order equilibrium finite element method, International Journal for Numeri- cal Methods in Engineering 114 (12) (2018) 1262–1290.doi:10.1002/ nme.5785

    work page 2018

  32. [32]

    Orynyak, K

    I. Orynyak, K. Danylenko, Method of matched sections as a beam-like approach for plate analysis, Finite Elements in Analysis and Design 230 (2024) 104103.doi:10.1016/j.finel.2023.104103

    work page doi:10.1016/j.finel.2023.104103 2024

  33. [33]

    Orynyak, A

    I. Orynyak, A. Tsybulnyk, K. Danylenko, Spectral realization of the method of matched sections for thin-plate vibration, Archive of Applied Mechanics 95 (2025) 51.doi:10.1007/s00419-024-02755-7. 41

    work page doi:10.1007/s00419-024-02755-7 2025

  34. [34]

    Orynyak, A

    I. Orynyak, A. Tsybulnyk, K. Danylenko, A. Oryniak, S. Radchenko, Timestep-dependent element interpolation functions in the method of matched sections on the example of heat conduction problem, Journal of Computational and Applied Mathematics 456 (2025) 116222.doi: 10.1016/j.cam.2024.116222

    work page doi:10.1016/j.cam.2024.116222 2025

  35. [35]

    Danylenko, I

    K. Danylenko, I. Orynyak, Numeric analysis of elastic plane body static problem by the method of matched sections, Mechanics and Advanced Technologies 8 (4(103)) (2024) 428–440.doi:10.20535/2521-1943. 2024.8.4(103).313412

    work page doi:10.20535/2521-1943 2024

  36. [36]

    Leckie, E

    F. Leckie, E. Pestel, Transfer-matrix fundamentals, International Jour- nal of Mechanical Sciences 2 (3) (1960) 137–167.doi:10.1016/ 0020-7403(60)90001-1

    work page 1960

  37. [37]

    Salomon, Curves and Surfaces for Computer Graphics, Springer New York, New York, 2005

    D. Salomon, Curves and Surfaces for Computer Graphics, Springer New York, New York, 2005

    work page 2005

  38. [38]

    R. Li, B. Wang, G. Li, Benchmark bending solutions of rectangular thin plates point-supported at two adjacent corners, Applied Mathematics Letters 40 (2015) 53–58.doi:10.1016/j.aml.2014.09.012

    work page doi:10.1016/j.aml.2014.09.012 2015

  39. [39]

    Y. Yi, Z. Tang, C. Liu, Progressive iterative approximation for extended B-Spline interpolation surfaces, Mathematical Problems in Engineering 2021 (2021) 5556771.doi:10.1155/2021/5556771

    work page doi:10.1155/2021/5556771 2021

  40. [40]

    Franke, A critical comparison of some methods for interpolation of scattered data, Tech

    R. Franke, A critical comparison of some methods for interpolation of scattered data, Tech. Rep. NPS-53-79-003, Naval Postgraduate School (1975). 42

    work page 1975