arXiv subjects

Jørgen S. Dokken

Publications and source records attributed to Jørgen S. Dokken.

12 recordsLinked to original sources

arXiv2026

RFBniCS: An open-source simulation framework for redox flow batteries

We present RFBniCS, an open-source finite-element framework implemented in FEniCSx for simulating redox-flow-battery (RFB) half-cells. RFBniCS solves an established macro-homogeneous porous-electrode model accounting for strongly coupled electrolyte flow, multicomponent species transport, ionic and electronic charge conservation, and interfacial Faradaic charge transfer. Different from other open-access tools, RFBniCS can perform transient RFB half-cell simulations in one-, two-, and three-dimensional geometries while explicitly resolving the transport of both redox-active and supporting-electrolyte species. It also describes electrolyte flow through porous electrodes and, in the three-dimensional formulation, through adjacent flow channels. To verify the implementation, RFBniCS's predictions in parameter-limiting regimes are compared against other published implementations, with RFBniCS generally showing superior accuracy and speed. We further demonstrate RFBniCS's capabilities by a simulation of the transient response of the negative half-cell of a vanadium redox flow battery, with a moderately concentrated supporting electrolyte. In the current implementation, RFBniCS provides the computational basis for studying redox-flow-battery half-cells and can be extended to complex flow-field designs, alternative chemistries, full-cell coupling, detailed membrane transport, and additional multiphysics effects.

physics.comp-ph↗

arXiv2026

Multiphysics tritium transport modelling of the ARC breeding blanket with FESTIM

Accurate prediction of tritium behaviour in molten salt breeding blankets is essential for the design and safe operation of ARC-class fusion reactors. This work presents a fully open-source, component-scale multiphysics framework for modelling tritium transport in an ARC liquid immersion blanket. Neutron transport, thermal hydraulics, and hydrogen isotope transport are coupled using OpenMC, OpenFOAM, and FESTIM, leveraging dedicated tools enabling direct transfer of spatially resolved fields between solvers. Assuming a zero inlet concentration, steady-state simulations predict a total tritium inventory of approximately 243 mg, with the blanket reaching steady-state tritium throughput within approximately 30 min, which is of a similar order to previous system-level estimates. The results show that tritium transport is dominated by turbulence-enhanced diffusion, with strong localisation in flow stagnation regions and reduced accumulation in highly turbulent zones. Sensitivity analyses indicate that predicted inventories are governed primarily by the numerical stabilisation scheme, with only a modest dependence on the turbulent Schmidt number. The proposed workflow provides a transparent and extensible basis for high-fidelity analysis of tritium transport in ARC-class breeding blankets.

physics.comp-ph↗

arXiv2026

Proximal Galerkin for Phase Field Fracture

The phase-field method has emerged as a powerful tool for simulating fracture mechanics, yet it presents significant numerical challenges, particularly regarding the enforcement of physical constraints such as irreversibility and boundedness of the phase-field variable. This work proposes the proximal Galerkin (PG) methodology as a robust and efficient framework for solving phase-field fracture problems. By reformulating the inequality-constrained optimization problem into a sequence of saddle-point problems involving latent variables, the PG method rigorously enforces the physical bounds of the phase-field variable and naturally handles the irreversibility condition. This approach is directly applicable to both static and dynamic phase-field fracture problems. The numerical results demonstrate that the PG framework accurately reproduces theoretical predictions and experimental observations, while offering a unified, mathematically consistent treatment of the constraints inherent to phase-field fracture modeling.

math.NA↗

arXiv2025

DefElement: an encyclopedia of finite element definitions

DefElement is an online encyclopedia of finite element definitions that was created and is maintained by the authors of this paper. DefElement aims to make information about elements defined in the literature easily available in a standard format. There are a number of open-source finite element libraries available, and it can be difficult to check that an implementation of an element in a library matches the element's definition in the literature or implementation in another library, especially when many libraries include variants of elements whose basis functions do not match exactly. In this paper, we carefully derive conditions under which elements can be considered equivalent and describe an algorithm that uses these conditions to verify that two implementations of a finite element are indeed variants of the same element. The results of scheduled runs of our implementation of this verification algorithm are included in the information available on the DefElement website.

math.NA↗

arXiv2025

FESTIM v2.0: Upgraded framework for multi-species hydrogen transport and enhanced performance

FESTIM is an open-source finite element framework for modelling the transport of hydrogen isotopes in materials. It provides a flexible and extensible tool for simulating diffusion, trapping, surface interactions, and other processes that govern hydrogen behaviour. This paper presents FESTIM v2.0, a major release that broadens both the physical scope and the software infrastructure of the framework. On the physics side, the formulation adopts a modular structure that supports multi-species transport, advanced trapping and reaction schemes, isotope exchange, decay, and advection. Interface and boundary conditions have been generalised, and interoperability with external solvers enables multiphysics workflows, including coupling with fluid dynamics and neutron transport codes. On the software side, FESTIM v2.0 has been migrated to DOLFINx, the next-generation FEniCS platform, providing improved performance, interoperability, and long-term sustainability. Taken together, these advances position FESTIM v2.0 as a versatile platform for investigating hydrogen transport in materials across scientific and engineering applications.

physics.comp-ph↗

arXiv2025

The latent variable proximal point algorithm for variational problems with inequality constraints

The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge--Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. In this paper, we describe the LVPP algorithm in a general form and apply it to ten problems from across mathematics.

math.OC↗

arXiv2024

Spatial modeling algorithms for reactions and transport (SMART) in biological cells

Biological cells rely on precise spatiotemporal coordination of biochemical reactions to control their many functions. Such cell signaling networks have been a common focus for mathematical models, but they remain challenging to simulate, particularly in realistic cell geometries. Herein, we present our software, Spatial Modeling Algorithms for Reactions and Transport (SMART), a package that takes in high-level user specifications about cell signaling networks and molecular transport, and then assembles and solves the associated mathematical and computational systems. SMART uses state-of-the-art finite element analysis, via the FEniCS Project software, to efficiently and accurately resolve cell signaling events over discretized cellular and subcellular geometries. We demonstrate its application to several different biological systems, including YAP/TAZ mechanotransduction, calcium signaling in neurons and cardiomyocytes, and ATP generation in mitochondria. Throughout, we utilize experimentally-derived realistic cellular geometries represented by well-conditioned tetrahedral meshes. These scenarios demonstrate the applicability, flexibility, accuracy and efficiency of SMART across a range of temporal and spatial scales.

q-bio.QM↗

arXiv2023

SMART: Spatial Modeling Algorithms for Reaction and Transport

Recent advances in microscopy and 3D reconstruction methods have allowed for characterization of cellular morphology in unprecedented detail, including the irregular geometries of intracellular subcompartments such as membrane-bound organelles. These geometries are now compatible with predictive modeling of cellular function. Biological cells respond to stimuli through sequences of chemical reactions generally referred to as cell signaling pathways. The propagation and reaction of chemical substances in cell signaling pathways can be represented by coupled nonlinear systems of reaction-transport equations. These reaction pathways include numerous chemical species that react across boundaries or interfaces (e.g., the cell membrane and membranes of organelles within the cell) and domains (e.g., the bulk cell volume and the interior of organelles). Such systems of multi-dimensional partial differential equations (PDEs) are notoriously difficult to solve because of their high dimensionality, non-linearities, strong coupling, stiffness, and potential instabilities. In this work, we describe Spatial Modeling Algorithms for Reactions and Transport (SMART), a high-performance finite-element-based simulation package for model specification and numerical simulation of spatially-varying reaction-transport processes. SMART is based on the FEniCS finite element library, provides a symbolic representation framework for specifying reaction pathways, and supports geometries in 2D and 3D including large and irregular cell geometries obtained from modern ultrastructural characterization methods.

q-bio.QM↗

arXiv2022

Model order reduction of solidification problems

Advection driven problems are known to be difficult to model with a reduced basis because of a slow decay of the Kolmogorov $N$-width. This paper investigates how this challenge transfers to the context of solidification problems and tries to answer when and to what extend reduced order models (ROMs) work for solidification problems. In solidification problems, the challenge is not the advection per se, but rather a moving solidification front. This paper studies reduced spaces for 1D step functions that move in time, which can either be seen as advection of a quantity or as a moving solidification front. Furthermore, the reduced space of a 2D solidification test case is compared with the reduced space of an alloy solidification featuring a mushy zone. The results show that not only the PDE itself, but the smoothness of the solution is crucial for the decay of the singular values and thus the quality of a reduced space representation.

math.NA↗

arXiv2022

Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes

We develop a method for generating degree-of-freedom maps for arbitrary order finite element spaces for any cell shape. The approach is based on the composition of permutations and transformations by cell sub-entity. Current approaches to generating degree-of-freedom maps for arbitrary order problems typically rely on a consistent orientation of cell entities that permits the definition of a common local coordinate system on shared edges and faces. However, while orientation of a mesh is straightforward for simplex cells and is a local operation, it is not a strictly local operation for quadrilateral cells and in the case of hexahedral cells not all meshes are orientable. The permutation and transformation approach is developed for a range of element types, including Lagrange, and divergence- and curl-conforming elements, and for a range of cell shapes. The approach is local and can be applied to cells of any shape, including general polytopes and meshes with mixed cell types. A number of examples are presented and the developed approach has been implemented in an open-source finite element library.

math.NA↗

arXiv2020

Automatic shape derivatives for transient PDEs in FEniCS and Firedrake

In industry, shape optimization problems are of utter importance when designing structures such as aircraft, automobiles and turbines. For many of these applications, the structure changes over time, with a prescribed or non-prescribed movement. Therefore, it is important to capture these features in simulations when optimizing the design of the structure. Using gradient based algorithms, deriving the shape derivative manually can become very complex and error prone, especially in the case of time-dependent non-linear partial differential equations. To ease this burden, we present a high-level algorithmic differentiation tool that automatically computes first and second order shape derivatives for partial differential equations posed in the finite element frameworks FEniCS and Firedrake. The first order shape derivatives are computed using the adjoint method, while the second order shape derivatives are computed using a combination of the tangent linear method and the adjoint method. The adjoint and tangent linear equations are symbolically derived for any sequence of variational forms. As a consequence our methodology works for a wide range of PDE problems and is discretely consistent. We illustrate the generality of our framework by presenting several examples, spanning the range of linear, non-linear and time-dependent PDEs for both stationary and transient domains.

math.OC↗

arXiv2019

A multimesh finite element method for the Navier-Stokes equations based on projection methods

The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche's method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable variational formulation. In this contribution we extend the multimesh finite element method to the Navier-Stokes equations based on the incremental pressure correction scheme. For each step in the pressure correction scheme, we derive a multimesh finite element formulation with suitable stabilization terms. The overall scheme yields expected spatial and temporal convergence rates on the Taylor-Green problem, and demonstrates good agreement for the drag and lift coefficients on the Turek-Schafer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a channel.

math.NA↗