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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +7359 +10.21105/joss.07359 + +ngsPETSc: A coupling between NETGEN/NGSolve and +PETSc + + + +https://orcid.org/0000-0002-3919-8603 + +Betteridge +Jack + + + + +https://orcid.org/0000-0002-1241-7060 + +Farrell +Patrick E. + + + + +https://orcid.org/0009-0001-8842-3221 + +Hochsteger +Matthias + + + + +https://orcid.org/0009-0000-3448-3002 + +Lackner +Christopher + + + + +https://orcid.org/0000-0002-1250-5087 + +Schöberl +Joachim + + + + +https://orcid.org/0000-0002-0435-0433 + +Zampini +Stefano + + + + +https://orcid.org/0000-0002-2577-1106 + +Zerbinati +Umberto + + +* + + + +Imperial College London, United Kingdom + + + + +CERBSim GmbH, Austria + + + + +University of Oxford, United Kingdom + + + + +TU Wien, Austria + + + + +King Abdullah University of Science and Technology, Saudi +Arabia + + + + +* E-mail: + + +1 +7 +2024 + +9 +104 +7359 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2024 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +PETSc +FEM +Meshing + + + + + + Summary +

Combining advanced meshing techniques with robust solver + capabilities is essential for solving difficult problems in + computational science and engineering. In recent years, various + software packages have been developed to support the integration of + meshing tools with finite element solvers. To mention a few, FreeFEM + (Hecht + et al., 2005) includes built-in support for mesh generation, + allowing users to create and manipulate meshes directly within the + software. Similarly, deal.II + (Arndt + et al., 2023) provides a GridGenerator class for generating + standard mesh geometries like grids and cylinders. Furthermore, + deal.II can interface with OpenCASCADE + (OpenCASCADE + Technology, 2023) to refine existing grids while conforming to + the geometry provided. Other finite element libraries, such as + Firedrake + (Ham + et al., 2023), DUNE-FEM + (Dedner + et al., 2010), and FEniCSx + (Baratta + et al., 2023), rely on external tools like Gmsh + (Geuzaine + & Remacle, 2020) and Tetgen + (Si, + 2015) for mesh generation. This paper introduces ngsPETSc, + software built with petsc4py + (Dalcin + et al., 2011) that seamlessly integrates the NETGEN mesher + (Schöberl, + 1997), the NGSolve finite element library + (Schöberl, + 2014), and the PETSc toolkit + (Balay + et al., 2023). ngsPETSc enables the use of NETGEN meshes and + geometries in solvers that use PETSc’s DMPLEX + (Lange + et al., 2015), and provides NGSolve users access to the wide + array of linear, nonlinear solvers, and time-steppers available in + PETSc.

+ + + Statement of Need +

Efficiently solving large-scale partial differential equations + (PDEs) on complex geometries is vital in scientific computing. PETSc, + NETGEN, and NGSolve offer distinct functionalities: PETSc handles + linear and nonlinear problems in a discretisation agnostic manner, + NETGEN constructs meshes from constructive solid geometry (CSG) + described with OpenCASCADE + (OpenCASCADE + Technology, 2023), and NGSolve offers a wide range of finite + element discretisations. Integrating these tools with ngsPETSc + promises to streamline simulation workflows and to enhance large-scale + computing capabilities for challenging problems. This integration also + facilitates seamless mesh exports from NETGEN to PETSc DMPlex, + enabling simulations of complex geometries and supporting advanced + meshing techniques in other PETSc-based solvers that employ DMPLEX. We + illustrate this with the Firedrake finite element system + (Ham + et al., 2023).

+

In particular, by combining PETSc, NETGEN, and NGSolve within + ngsPETSc the following new features are available:

+ + +

PETSc Krylov solvers, including flexible and pipelined + variants, are available in NGSolve. They can be used both with + NGSolve matrix-free operators and NGSolve block matrices;

+ + +

PETSc preconditioners can be used as components within the + NGSolve preconditioning infrastructure;

+ + +

PETSc nonlinear solvers are available in NGSolve, including + advanced line search and trust region Newton-based methods;

+ + +

high order meshes constructed in NETGEN are now available in + Firedrake + (Ham + et al., 2023), enabling adaptive mesh refinement and + geometric multigrid on hierarchies of curved meshes.

+ + +

In conclusion, ngsPETSc is a lightweight, user-friendly interface + that bridges the gap between NETGEN, NGSolve, and PETSc, building on + top of petsc4py. ngsPETsc aims to assist with the solution of + challenging PDEs on complex geometries, enriching the already powerful + capabilities of NETGEN, NGSolve, PETSc, and Firedrake.

+ + + Examples +

In this section we provide a few examples of results that can be + obtained using ngsPETSc. We begin by considering a simple primal + Poisson problem on a unit square domain discretised with conforming + + + P2 + finite elements and compare the performance of different solvers newly + available in NGSolve via ngsPETSc. In particular, we consider PETSc’s + algebraic multigrid algorithm GAMG + (“Parallel + Multigrid Smoothing,” 2003), PETSc’s domain decomposition BDDC + algorithm + (Zampini, + 2016), NGSolve’s own implementation of element-wise BDDC, the + Hypre algebraic multigrid algorithm + (Falgout + & Yang, 2002) and the ML algebraic multigrid algorithm + (Sala et + al., 2004), each combined with the conjugate gradient method. + Other than the elementwise BDDC preconditioner, these preconditioners + were not previously available in NGSolve. The results are shown in + Table 1 and the full example, with more details, can be found in the + ngsPETSc + documentation. All the preconditioners considered exhibit + robust conjugate gradient iteration counts as we refine the mesh for a + + + P1 + discretisation, but out-of-the-box only BDDC type preconditioners are + robust as we refine the mesh for a + + P2 + discretisation. A possible remedy for this issue is discussed in the + ngsPETSc + documentation.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
# DoFsPETSc GAMGHYPREMLPETSc BDDC*Element-wise BDDC**
116716353631910
46485869746389
1855428142148127910
+ +

Table 1: The number of degrees of freedom (DoFs) and the number of + iterations required to solve the Poisson problem with different + solvers. Each row corresponds to a level of uniform refinement. The + conjugate gradient solve was terminated when the residual norm + decreased by six orders of magnitude. *We choose to use PETSc BDDC + with six subdomains. **Element-wise BDDC is a custom implementation of + BDDC in NGSolve.

+

We next consider the Oseen problem, i.e. + We discretise this problem using + high-order Hood-Taylor elements ( + + P4- + + P3) + on a unit square domain + (Boffi, + 1994; + Taylor + & Hood, 1973). We employ an augmented Lagrangian + formulation to better enforce the incompressibility constraint. We + present the performance of GMRES + (Saad + & Schultz, 1986) preconditioned with a two level additive + Schwarz preconditioner with vertex-patch smoothing as fine level + correction + (Benzi + & Olshanskii, 2006; + Farrell + et al., 2021). This preconditioner was built using ngsPETSc. + The result for different viscosities + + ν + are shown in Table 2, exhibiting reasonable robustness as the + viscosity (and hence Reynolds number) changes. The full example, with + more details, can be found in the + ngsPETSc + documentation.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
# refinements (# DoFs) + + ν=10−2 + + ν=10−3 + + ν=10−4
1 (83842)346
2 (334082)346
3 (1333762)346
+ +

Table 2: The number of iterations required to solve the Oseen + problem with different viscosities and different refinement levels. In + parentheses we report the number of degrees of freedom (DoFs) on the + finest level. The GMRES iteration was terminated when the residual + norm decreased by eight orders of magnitude.

+

Figure 1 shows a simulation of a hyperelastic beam, solved with + PETSc nonlinear solvers; the line search algorithms in PETSc solve + this straightforwardly, but an undamped Newton iteration does not + converge. Figures 2 and 3 show simulations in Firedrake that were not + previously possible. Figure 2 shows a high-order NETGEN mesh employed + for the simulation of a Navier-Stokes flow past a cylinder, while + Figure 3 shows adaptive mesh refinement for a Poisson problem on an + L-shaped domain. The adaptive procedure achieves the optimal + complexity of error with degree of freedom count, as expected + (Stevenson, + 2006).

+ +

A hyperelastic beam deformed by fixing one end and + applying a twist at the other end. The colouring corresponds to the + deviatoric von Mises stress experienced by the beam. The beam is + discretised with + + P3 + finite elements and the nonlinear problem is solved using PETSc + SNES. The full example, with more details, can be found in the + ngsPETSc + documentation.

+ + + +

Flow past a cylinder. The Navier-Stokes equations are + discretised on a NETGEN high-order mesh with Firedrake. We use + high-order Taylor-Hood elements ( + + P4- + + P3) + and a vertex-patch smoother as fine level correction in a two-level + additive Schwarz preconditioner, + (Benzi + & Olshanskii, 2006; + Farrell + et al., 2021). The full example, with more details, can be + found in + ngsPETSc + documentation. On the right a zoom near the cylinder + shows the curvature of the mesh.

+ + + +

An adaptive scheme applied to the Poisson problem on an + L-shaped domain. The domain is discretised using + + + P1 + finite elements and the adaptive mesh refinement is driven by a + Babuška-Rheinboldt error estimator + (Babuška + & Rheinboldt, 1978). The adaptive procedure delivers + optimal scaling of the energy norm of the error in terms of the + number of degrees of freedom. The full example, with more details, + can be found in the + ngsPETSc + documentation.

+ + +

More examples can be found in the documentation of ngsPETSc manual + (Zerbinati, + 2024).

+ + + Acknowledgement +

This work was funded by the Engineering and Physical Sciences + Research Council [grant numbers EP/R029423/1 and EP/W026163/1]. For + the purpose of open access, the author has applied a CC BY public + copyright licence to any Author Accepted Manuscript (AAM) version + arising from this submission.

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