arXiv subjects

André Massing

Publications and source records attributed to André Massing.

15 recordsLinked to original sources

arXiv2026

A cut finite element method for the Biot system of poroelasticity

We propose a novel cut finite element method for the numerical solution of the Biot system of poroelasticity. The Biot system couples elastic deformation of a porous solid with viscous fluid flow and commonly arises on domains with complex geometries that make high-quality volumetric meshing challenging. To address this issue, we employ the cut finite element framework, where the domain boundary is represented independently of the background mesh, which significantly simplifies the meshing process. Our approach builds upon a parameter robust total pressure formulation of the Biot system, which we combine with the cut finite element method to develop a geometrically robust solution scheme, while preserving the parameter robustness. A key ingredient in the theoretical analysis is a modified inf-sup condition which also holds for mixed boundary conditions, leading to stability and optimal error estimates for the proposed formulation. Finally, we provide numerical evidence demonstrating the theoretical properties of the method and showcasing its capabilities by solving the Biot system on a realistic brain geometry.

math.NA↗

arXiv2026

A finite element framework for solving coupled multiphysics problem with moving boundaries in cell biophysics

Cellular morphodynamics requires solving systems of coupled partial differential equations on moving bulk and surface domains, where advection-dominant transport, structure preservation, and severe mesh distortions make robust simulation difficult. We present a holistic finite element framework that jointly addresses these obstacles for biophysical applications by combining model-agnostic structure-preserving postprocessing, ALE-based mesh redistribution strategies driven by surface-tangential velocities, and stabilized discretization for advection-diffusion-reaction problems tailored to evolving domains. The methodology is modular and applies to advection-diffusion-reaction systems, Cahn-Hilliard phase separation, Helfrich-type geometric flows, as well as their staggered and potentially mixed-dimensional couplings. We provide a concise notation for evolving bulk and surface geometries, extend positivity-, bound-, and mass-preserving projections to moving meshes, and develop a two-step redistribution procedure that maintains element quality without remeshing. Convergence studies, manufactured solutions, and biologically motivated test cases -- including tumor-growth surrogates and phase segregation on deformable membranes -- demonstrate accuracy, stability, and versatility across the problem classes considered.

math.NA↗

arXiv2025

Thermal performance estimation for cryogenic storage tanks: Application to liquid hydrogen

The design of cryogenic liquid storage solutions requires accurate methods for estimating heat ingress, from the material level to the tank level. For insulation materials, thermal performance is usually measured using ambient conditions and liquid nitrogen at 77 K as boundary temperatures. A key question is how much heat ingress increases when storing liquid hydrogen LH$_2$ at 20 K. We derive theoretical bounds on the increased heat ingress, and show that it remains below 26%. Additionally, we demonstrate that heat ingress is much more sensitive to the warm boundary temperature than the cold boundary temperature. At the tank level, we compare two methods for assessing the steady-state thermal performance of cryogenic tanks: thermal network models and the heat equation solved with the finite element method. The latter offers high accuracy and adaptability for complex geometries, while thermal network models benefit from simplicity, speed and robustness. We apply both approaches to a self-supported LH$_2$ tank concept for maritime transport and analyze sensitivity to structural support thickness, warm boundary temperature, and choice of insulation material. The thermal network model can estimate heat ingress with $\sim$1% error and the cold-spot temperature with error less than 1 K.

physics.app-ph↗

arXiv2024

Local Area Cooling versus Broad Area Cooling for Boil-Off Reduction in Large-Scale Liquid Hydrogen Storage Tanks

Future use of liquid hydrogen (LH$_2$) as an effective energy carrier will require elimination or minimization of hydrogen boil-off that is not utilized by demands in the value chain. The present work promotes local area cooling (LAC) as a promising boil-off reduction technology. In contrast to the more conventional broad area cooling (BAC), LAC targets local, concentrated heat flows e.g. through tank support structures. This yields important practical benefits, especially for large-scale tanks, due to the order-of-magnitude reduction in the size of the cooling system. Such benefits include lower capital costs and simpler installation, maintenance and coolant management. LAC applied outside the outer tank wall is particularly attractive for tanks with evacuated insulation. In a series of numerical studies, we use the finite element method to evaluate the thermal performance of LAC and BAC in the context of ship-borne LH$_2$ transport. The studies concern 40 000 m$^3$-capacity, skirt-supported tanks insulated using evacuated perlite or helium-filled polyurethane (HePUR) foam. For the perlite-insulated tank, LAC and BAC with liquid nitrogen coolant can reduce the daily boiloff rate from 0.04%/day to, respectively, 0.011%/day and 0.004%/day. The corresponding numbers for CO$_2$-based refrigeration are 0.031%/day and 0.028%/day. For the HePUR-insulated tank, which has a higher baseline boil-off rate of 0.24%/day, reduced boil-off rates down to 0.17%/day and 0.04%/day are achievable using LAC and BAC, respectively. LAC and BAC both offer increased power efficiency in comparison to reliquefaction only.

physics.ins-det↗

arXiv2024

A better compression driver? CutFEM 3D shape optimization taking viscothermal losses into account

The compression driver, the standard sound source for midrange acoustic horns, contains a cylindrical compression chamber connected to the horn throat through a system of channels known as a phase plug. The main challenge in the design of the phase plug is to avoid resonance and interference phenomena. The complexity of these phenomena makes it difficult to carry out this design task manually, particularly when the phase-plug channels are radially oriented. Therefore, we employ an algorithmic technique that combines numerical solutions of the governing equations with a gradient-based optimization algorithm that can deform the walls of the phase plug. A particular modeling challenge here is that viscothermal losses cannot be ignored, due to narrow chambers and slits in the device. Fortunately, a recently developed, accurate, but computationally inexpensive boundary-layer model is applicable. We use this model, a level-set geometry description, and the Cut Finite Element technique to avoid mesh changes when the geometry is modified by the optimization algorithm. Moreover, the shape calculus needed to compute derivatives for the optimization algorithm is carried out in the fully discrete case. Applying these techniques, the algorithm was able to successfully design the shape of a set of radially-directed phase plugs so that the final frequency response surprisingly closely matches the ideal response, derived by a lumped circuit model where wave interference effects are not accounted for. This result may serve to resuscitate the radial phase plug design, rarely used in today's commercial compression drivers.

math.NA↗

arXiv2023

Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates

We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.

math.NA↗

arXiv2023

Stabilized cut discontinuous Galerkin methods for advection-reaction problems on surfaces

We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection-reaction problems on surfaces embedded in $\mathbb{R}^d$. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order $k$ as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.

math.NA↗

arXiv2020

Stabilized CutDG methods for advection-reaction problems

We develop novel stabilized cut discontinuous Galerkin (CutDG) methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in $\mathbb{R}^d$ where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi)-norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two and three-dimensional test problems.

math.NA↗

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↗

arXiv2018

A stabilized cut discontinuous Galerkin framework: I. Elliptic boundary value and interface problems

We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of el- liptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the back- ground mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and con- dition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems.

math.NA↗

arXiv2017

A note on the penalty parameter in Nitsche's method for unfitted boundary value problems

Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. From the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors.

math.NA↗

arXiv2016

Full Gradient Stabilized Cut Finite Element Methods for Surface Partial Differential Equations

We propose and analyze a new stabilized cut finite element method for the Laplace-Beltrami operator on a closed surface. The new stabilization term provides control of the full $\mathbb{R}^3$ gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace-Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and $L^2$ norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

math.NA↗

arXiv2015

A Stabilized Cut Finite Element Method for the Three Field Stokes Problem

We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure and extra-stress tensor are discretised on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings with numerical examples.

math.NA↗

arXiv2012

Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D

In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on non-matching or overlapping meshes. Examples of such methods include the fictitious domain method, the extended finite element method, and Nitsche's method. In all of these methods, integrals must be computed over cut cells or subsimplices which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsche's method which we apply to Poisson's equation and a linear elastic problem.

math.NA↗

arXiv2012

A stabilized Nitsche overlapping mesh method for the Stokes problem

We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By ex- tending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.

math.NA↗