Associated Project AP10 • Numerical methods for elastic wave propagation in heterogeneous spatial networks

Motivation

Many problems in science and engineering are governed by partial differential equations (PDEs) on geometrically complex domains, which gives rise to significant computational challenges. For instance, when the domain is locally slender, classical simulation techniques such as the finite element method require excessive spatial resolution and can produce nearly degenerate elements. As a remedy, one may employ spatial network models, which encode the locally one-dimensional geometry of the underlying domain as a graph, where the edges correspond to slender segments of the domain and the nodes represent their connections. The local behavior of the problem is then described by one-dimensional differential equations on the edges which are suitably coupled at the network nodes. Compared to the underlying full three-dimensional model, spatial network models can substantially reduce the computational complexity in a numerical simulation, while preserving the essential characteristics of the problem. A prominent application of spatial network models, is fiber-based materials, which appear in a wide range of engineering applications, including paper, nonwoven fabrics, and sintered metallic fiber networks. Here, the deformation of fiber segments can be modeled using one-dimensional beam models that are suitably coupled at nodes representing the contact regions of fibers [KU12, S20]. Other important applications include the modeling of porous media, where the detailed pore space is approximated by a network of pore cavities connected by throats [BBDGIMPP12], and biological systems, such as vascular networks, where blood vessels can be modeled as connected one-dimensional edges [KVW20]. Illustrations of these applications are shown in Figure 1.

           

The applications of spatial network models mentioned above are inherently heterogeneous in nature, where the heterogeneity is caused by the complex geometric structure of the problems and highly varying material coefficients. As a result, the effective behavior of such spatial network models is determined by the interaction of effects at multiple length scales, and therefore they can be classified as multiscale problems. Note that although spatial network models are already simplifications of more complex three-dimensional problems, they may still be intractable using standard discretization methods that resolve all scales globally.

Aims of the project

The overall goal of this project is to address the numerically challenging problem of elastic wave propagation in heterogeneous spatial network models. To this end, we design, analyze, and implement efficient numerical methods that overcome the severe stability and conditioning issues inherent to such models. In previous work, we have developed computational multiscale methods for spatial networks [HM24, HMM26] as well as efficient two-level domain decomposition techniques [HMR25]. A cornerstone for the analysis of these numerical methods is a rigorous theoretical framework [CY95, GHM23] that enables, at sufficiently coarse discretization scales, the transfer of fundamental results from the theory of partial differential equations in the continuous setting, such as Poincaré-type inequalities, to discrete spatial network models. This framework makes it possible to rigorously analyze numerical methods utilizing coarse discretization scales, which are an essential feature of efficient numerical schemes.

For wave propagation problems, the highly heterogeneous network structure leads to particularly restrictive CFL conditions for explicit time stepping schemes, caused by the presence of very short fibers. Implicit time integration methods alleviate these stability constraints but instead give rise to severely ill-conditioned linear systems that must be solved at each time step.

To overcome these challenges, we develop numerical methods that combine techniques from domain decomposition, computational multiscale methods, and locally implicit time stepping schemes [BH25]. The proposed methods aim to relax global stability restrictions and overcome conditioning issues of the resulting linear systems through suitable preconditioning techniques, enabling efficient numerical simulations of elastic waves on spatial network models.

Preprints

1. M. Hauck, C. Ma, and D. Peterseim. Local approximation of $A$-harmonic functions and the super-localization conjecture. CRC 1173 Preprint 2026/54, Karlsruhe Institute of Technology, October 2026. [bibtex]

2. M. Hauck, J. Holten, A. Målqvist, A. Rupp, and L. Swoboda. A hybridizable discontinuous Galerkin method for wave propagation in elastic beam networks. CRC 1173 Preprint 2026/46, Karlsruhe Institute of Technology, August 2026. [files] [bibtex]

3. M. Hauck, R. Maier, and T. Sprekeler. A post-processed higher-order multiscale method for nondivergence-form elliptic equations. CRC 1173 Preprint 2026/13, Karlsruhe Institute of Technology, April 2026. [files] [bibtex]

Other references

4. T. Buchholz and M. Hochbruck. A non-iterative domain decomposition time integrator for linear wave equations. Math. Comp., 1–30, August 2026. URL https://doi.org/10.1090/mcom/4234. Online first. [preprint] [files] [bibtex]

5. M. Hauck, A. Målqvist, and A. Rupp. Arbitrary order approximations at constant cost for Timoshenko beam network models. ESAIM M2AN, 59(6):3107–3130, December 2025. URL https://doi.org/10.1051/m2an/2025085. [bibtex]

6. M. Hauck and A. Målqvist. Super-localization of spatial network models. Numer. Math., 156(3):901–926, June 2024. URL https://doi.org/10.1007/s00211-024-01410-1. [bibtex]

7. M. Görtz, F. Hellman, and A. Målqvist. Iterative solution of spatial network models by subspace decomposition. Math. Comp., 93(345):233–258, January 2024. URL https://doi.org/10.1090/mcom/3861. [bibtex]

8. M. Hauck, R. Maier, and A. Målqvist. An algebraic multiscale method for spatial network models. arXiv preprint, Karlsruhe Institute of Technology, December 2023. [bibtex]

9. J.-W. Simon. A review of recent trends and challenges in computational modeling of paper and paperboard at different scales. Arch. Computat. Methods Eng., 28(4):2409–2428, June 2021. URL https://doi.org/10.1007/s11831-020-09460-y. [bibtex]

10. T. Köppl, E. Vidotto, and B. Wohlmuth. A 3D-1D coupled blood flow and oxygen transport model to generate microvascular networks. Int. J. Numer. Meth. Biomed. Engng., 36(10):e3386, October 2020. URL https://doi.org/10.1002/cnm.3386. [bibtex]

11. Z. Yi, M. Lin, W. Jiang, Z. Zhang, H. Li, and J. Gao. Pore network extraction from pore space images of various porous media systems. Water Resour. Res., 53(4):3424–3445, April 2017. URL https://doi.org/10.1002/2016WR019272. [bibtex]

12. M. J. Blunt, B. Bijeljic, H. Dong, O. Gharbi, S. Iglauer, P. Mostaghimi, A. Paluszny, and C. Pentland. Pore-scale imaging and modelling. Adv. Water Resour., 51(8):197–216, January 2013. URL https://doi.org/10.1016/j.advwatres.2012.03.003. 35th Year Anniversary Issue. [bibtex]

13. A. Kulachenko and T. Uesaka. Direct simulations of fiber network deformation and failure. Mech. Mater., 51(8):1–14, August 2012. URL https://doi.org/10.1016/j.mechmat.2012.03.010. [bibtex]

14. F. R. K. Chung and S.-T. Yau. Eigenvalues of graphs and Sobolev inequalities. Combin. Probab. Comput., 4(1):11–25, March 1995. URL https://doi.org/10.1017/S0963548300001449. [bibtex]