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A discontinuous Petrov-Galerkin finite-element framework for the simulation of microwave-heated flows
Oreste Marquis, Matthias Maier, Bruno Blais
TL;DR
Microwave-heated flow simulation requires accurate electromagnetic modeling coupled to thermal and fluid dynamics. The paper develops a high-order DPG-Maxwell solver within a multiphysics framework, validates it on analytical, numerical, and experimental benchmarks, and demonstrates adaptive and obstacle-flow simulations while identifying preconditioning and computational-cost extensions as future work.
Problem
Microwave-assisted chemical processes are difficult to model because electromagnetic fields depend sensitively on material properties and geometry, while high-wavenumber simulations increase computational cost.
Method
The paper couples a high-order DPG finite-element solver for time-harmonic Maxwell equations with continuous Galerkin heat-transfer and incompressible-flow solvers in Lethe.
Results
The electromagnetic solver reproduces expected waveguide convergence, agrees with Fichera-oven and obstacle benchmarks, and supports adaptive refinement and microwave-heated flow simulations.
Takeaways & Limitations
The framework provides a high-order, adaptive multiphysics approach for simulating microwave-heated flows and exploring how obstacle geometry affects heating and surrounding flow.
Takeaways & Limitations
The current implementation uses an unpreconditioned CG solver, while tailored preconditioning and sum-factorization are left for future work.
Abstract
from arXiv · showhide
We present a high-order multiphysics solver for the simulation of microwave-heated flows. The solver couples a discontinuous Petrov-Galerkin (DPG) finite element method for the time-harmonic Maxwell equations with continuous Galerkin finite element methods for the heat equation and the incompressible Navier-Stokes equations. We validate the electromagnetic solver against multiple benchmark problems: wave propagation in a rectangular waveguide, a cavity problem with a singular solution, and a microwave-heated obstacle problem, comparing our results against numerical and experimental data from the literature. The results confirm the validity of the implementation and demonstrate its ability to perform adaptive mesh refinement using the DPG method's built-in error estimator. The final part of the study showcases the capabilities of the multiphysics framework through simulations of microwave-heated flow around obstacles with singular geometric features. These results highlight the potential of the proposed framework for the simulation and optimization of microwave-assisted chemical processes. Finally, the developed high-order multiphysics solver has a low memory footprint, since the electromagnetic solver relies on a Conjugate Gradient (CG) iterative solver and the fluid solver is implemented in a matrix-free fashion, making the overall approach scalable and well-suited for large-scale parallel simulations.
1. Introduction
Microwave-assisted processes can reduce environmental impacts through selective volumetric heating, but their industrial adoption is constrained by electromagnetic sensitivity and simulation challenges. This work addresses these challenges by providing a practical DPG implementation and integrating it into an open-source multiphysics framework.
- The chemical industry’s approximately 5% share of global CO2 emissions motivates process innovation rather than simply reducing chemical production.
- Microwave-assisted processes provide selective, volumetric heating that can accelerate reactions, improve selectivity, and reduce undesired by-products.
- Industrial adoption remains limited because electromagnetic fields are sensitive to material properties and geometry, while internal field measurements are difficult.
- High-wavenumber simulations require suitable mesh size and polynomial degree, increasing degrees of freedom and making large-scale direct solvers impractical.
- DPG is selected because its Hermitian positive definite system enables Conjugate Gradient solvers and its built-in error estimator supports adaptive control.
- The work contributes a practical time-harmonic DPG implementation guide and integrates a DPG electromagnetic solver into Lethe for coupled flow, heat-transfer, and microwave simulations.
2. The DPG method for time-harmonic Maxwell equations
The framework applies an ultraweak DPG formulation to time-harmonic Maxwell equations using broken trial and enriched test spaces, trace unknowns, and locally computed residual information. It couples this electromagnetic discretization with boundary-aware field modeling, adjoint-graph-norm residual minimization, static condensation, and an unpreconditioned CG solve.
- 2.1. Abstract problem formulation: The practical DPG formulation uses discontinuous trial and enriched test spaces plus a mesh-skeleton trace space to weakly enforce interelement continuity.The discrete unknowns include the interior field, trace field, and error representation function.
- 2.1. Abstract problem formulation: The formulation enforces test-space orthogonality of the error representation function, with interior and trace unknowns acting as Lagrange multipliers.
- 2.2. Model problem and ultraweak formulation: The Maxwell model uses dimensionless time-harmonic electric and magnetic fields with material effects represented through effective relative permittivity and prescribed boundary data.
- 2.2. Model problem and ultraweak formulation: The ultraweak formulation retains both electric and magnetic field solutions, supporting computation of electromagnetic power dissipation in materials with potentially important magnetic effects.
- 2.3. Numerical implementation and discretization: An adjoint graph norm defines the residual minimization norm and is chosen so the broken test-space computation can be localized element by element.
- 2.3. Numerical implementation and discretization: Static condensation eliminates interior unknowns to form a Schur-complement system for trace unknowns, after which interior fields and the error representation function are recovered locally.
- 2.3. Numerical implementation and discretization: The condensed system is solved with an unpreconditioned CG method; tailored preconditioning remains future work, and unconstrained magnetic boundary degrees of freedom do not affect convergence for p≥1.
3. Multiphysics coupling
The framework extends lethe with electromagnetic, heat-transfer, and fluid-dynamics solvers for microwave-heated flows. It uses stabilized finite-element formulations and sequential electromagnetic–thermal–fluid coupling under a time-scale separation assumption.
- Multiphysics framework: lethe couples a DPG electromagnetic solver with existing heat-transfer and fluid-dynamics solvers to simulate microwave-heated flows.The integration supports independent discretization orders for each physics, enabling flexible computational-resource allocation.
- Fluid dynamics: The fluid model solves incompressible Newtonian Navier–Stokes equations with SUPG momentum stabilization and PSPG continuity stabilization based on local flow conditions and mesh size.The stabilizations address convection-related numerical instabilities and the saddle-point structure of the system.
- Heat transfer: The heat-transfer model solves enthalpy conservation, with electromagnetic power dissipation Q_em supplied as the volumetric heat source.For time-harmonic fields, Q_em is obtained from the time-averaged Poynting vector and material-loss contributions.
- Heat transfer: The enthalpy equation uses SUPG stabilization to mitigate numerical instabilities in high-Péclet-number flows.Temperature is represented through enthalpy as H = c_p T, with Dirichlet temperature conditions incorporated into the trial and test spaces.
- Coupling strategy: Time-scale separation justifies sequential coupling: electromagnetic fields are solved initially and when needed, while Q_em remains frozen between electromagnetic solves.Microwave electromagnetic oscillations occur on nanosecond periods, much faster than fluid or thermal changes, so the source evolves quasi-statically.
4. Numerical tests
The numerical tests verify the DPG electromagnetic solver on waveguide and singular-cavity benchmarks, then demonstrate coupled microwave-heated flow simulations around obstacles. Results show expected convergence, adaptive refinement near singularities, and geometry-dependent electromagnetic heating and flow responses.
- 4.1. Waveguide analytical verification: The waveguide solution recovers electromagnetic oscillations after sufficient refinement, with DPG localizing phase shifts within elements and exhibiting primarily amplitude attenuation under resolution limits.The observed behavior agrees with prior DPG wave-propagation studies and contrasts with larger phase errors reported for standard FEM formulations.
- 4.1. Waveguide analytical verification: Higher-order polynomials achieve the expected p+1 L2 convergence rate and can deliver converged wave solutions with fewer degrees of freedom than mesh refinement alone.The reported comparison includes p=3 on a coarser mesh versus p=1 on a more refined mesh.
- 4.2. Fichera oven: In the Fichera oven, the built-in error estimator refines cells near the re-entrant corner and edges, matching the locations of electromagnetic singularities.The resulting refinement pattern is qualitatively consistent with the reference study.
- 4.2. Fichera oven: The Fichera-oven global energy-norm error decreases monotonically with system degrees of freedom, confirming the implementation and h-adaptive refinement behavior.The convergence trend agrees with Petrides and Demkowicz despite the absence of a known analytical solution.
- 4.3. Flow around a microwave-heated body: The microwave-heated obstacle simulations converge with mesh refinement and polynomial order, while the computed field pattern agrees with prior results despite an input-power convention difference.The convention changes amplitude scaling but does not affect the reported solution pattern.
- 4.3. Flow around a microwave-heated body: Obstacle geometry changes heating: square prisms heat more than the cylinder, with about a 20 K average-temperature difference after 60 s and a tilted prism heating about 0.375 K s^-1 faster than the cylinder.The penetration depth is approximately 6 cm, comparable to the obstacle diameter, enabling volumetric absorption; equilibrium is not reached within the simulated interval.
5. Conclusions
The framework combines DPG electromagnetics with heat-transfer and fluid-dynamics solvers, achieving benchmark agreement, adaptive refinement, and scalable microwave-heated-flow simulations.
- 5. Conclusions: The integrated framework supports independently chosen high-order approximations and adaptive mesh refinement through the DPG method’s built-in error estimator.This enables different polynomial orders for each physical field and more efficient allocation of degrees of freedom.
- 5. Conclusions: The solver matched analytical, numerical, and experimental benchmarks for waveguide, Fichera oven, and microwave-heated obstacle problems.It reproduced convergence behavior, singularity-focused mesh adaptation, electric-field patterns, and temperature increases reported in prior work.
- 5. Conclusions: Simulations at Reynolds number 400 showed that obstacle geometry affects internal heating and the surrounding flow that extracts heat.The cases also demonstrated high temperatures in short times and multiphysics simulations using different polynomial orders.
- 5. Conclusions: The DPG-enabled conjugate-gradient electromagnetic solver handles large systems with a limited memory footprint for multiphysics simulations.The authors identify this as useful for the memory demands of large coupled problems.
- 5. Conclusions: Future work targets sum-factorization, iterative-solver preconditioning, and coupling to discrete-element simulations of microwave-heated multiphase flows.These directions address computational cost, solver convergence, and expanded multiphase capabilities.
A. Dimensionless conventions
The electromagnetic solver is formulated in dimensionless variables using reference scales for electric field, impedance, material properties, and length.
- A. Dimensionless conventions: The time-harmonic electromagnetic solver uses a dimensionless formulation based on explicit physical reference scales.The scaling conventions convert dimensional electromagnetic quantities into normalized variables for computation.
- A. Dimensionless conventions: The reference quantities include electric field, vacuum impedance, permittivity, permeability, speed of light, and problem length.The subscript "dim" identifies dimensional variables when notation could otherwise be ambiguous.
B. Polynomial spaces
The implementation constructs broken and trace finite-element spaces from standard Sobolev spaces on a three-dimensional hexahedral mesh.
- B. Polynomial spaces: The discretization uses broken trial and test spaces on mesh elements, while trace spaces represent interface quantities on the mesh skeleton.The broken spaces are defined element-wise, and trace operators establish the interface spaces needed by the formulation.
- B. Polynomial spaces: Standard Sobolev spaces provide the continuous starting point for defining the discretized and broken spaces.The stated H(curl, Ω) space requires both the field and its curl to belong to square-integrable vector spaces.
- B. Polynomial spaces: The three-dimensional implementation is restricted to discretized spaces over a hexahedral mesh and its element restrictions.The construction treats L2 functions as globally equivalent to their broken counterparts because they require no inter-element regularity.
- B. Polynomial spaces: The finite-element sequence uses trimmed tensor-product spaces for differential forms, including H(curl)-, H(div)-, and L2-conforming components.These correspond respectively to Nédélec, Raviart–Thomas, and discontinuous elements in the stated polynomial-space hierarchy.
C.1. Waveguide
The rectangular-waveguide verification records the solver configuration and linear-solver tolerance used for the benchmark simulation.
- C.1. Waveguide: Table 3 summarizes the parameters used for the rectangular-waveguide verification simulation.The table is the designated source for the benchmark setup values.
- C.1. Waveguide: The linear solver uses a relative residual tolerance of 10^-8.This tolerance is listed among the waveguide simulation parameters.
C.2. Fichera oven
The Fichera oven verification simulation is documented through a summarized set of solver parameters.
- The Fichera oven verification simulation's solver parameters are summarized in Table 4.
C.3. Microwave-heated flow
The microwave-heated-flow simulations document separate parameter summaries for cylinder cases without and with inlet velocity, alongside fluid-dynamics and heat-transfer solver settings.
- The cylinder simulation parameters are reported separately for cases without and with inlet velocity.
- The fluid-dynamics solver uses a GCMG preconditioner with 500 maximum Krylov vectors, three smoother iterations, and a direct coarse-grid solver.
- The heat-transfer solver uses GMRES with a relative residual tolerance of 10^-4.
CRediT authorship contribution statement
The authors divide contributions across conceptualization, methodology, software, validation, writing, and project leadership.
- Oreste Marquis led formal analysis, investigation, methodology, software, validation, visualization, and the original draft.
- Matthias Maier contributed to conceptualization, methodology, and review and editing.
- Bruno Blais contributed to conceptualization, funding acquisition, methodology, project administration, resources, software, supervision, and review and editing.