Free Boundary Value Problem Research Articles

Temperature field and residual stress distribution for laser metal deposition

Laser metal deposition (LMD) process has been widely used in many industrial applications such as automotive, defense, aerospace, and so on. Modeling has to address physical phenomena like laser-powder interaction, heat transfer, fusion, solidification, and track formation. LMD induces a complex thermal stress field that results in residual stress distributions. Depending on its magnitude and nature (i.e., whether tensile or compressive), the residual stresses can cause unpredicted in-service failures. Therefore, the prediction of its distribution in the deposited structure as a function of the process strategy is essential to improve the process and the part quality. LMD represents mathematically a free boundary value problem. This means that the track geometry is part of the solution. The authors developed a three-dimensional time-dependent finite element model for LMD with coaxial powder feeding supply. The model encompasses the powder stream, its interaction with the laser radiation, and the melt pool computation. The model was validated by a comparison of the experimental and computed shapes of the melt pool surfaces concerning cross section, longitudinal section and high-speed photographs [Pirch et al., in Proceedings of the ICALEO Conference, 16–20 October 2016]. In this paper, the model was applied to overlapping tracks for single and multilayer processing for different process strategies. The simulation allows us to analyze the time- and space-resolved evolution of temperature and stresses. The influence of the powder feed rate on the residual stresses is investigated.

Invasion moving boundary problem for a biofilm reactor model

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

A System Approach to Mathematical and Computer Modeling of Geomigration Processes Using Freefem++ and Parallelization of Computations

A method is described for constructing mathematical models of interrelated processes in porous media that are complex multicomponent systems. The performance capabilities of the package FreeFem++ are shown as applied to solving corresponding free boundary-value problems for systems of quasilinear parabolic equations by the finite element method using the parallelization of computations.

The island dynamics model on parallel quadtree grids

We introduce an approach for simulating epitaxial growth by use of an island dynamics model on a forest of quadtree grids, and in a parallel environment. To this end, we use a parallel framework introduced in the context of the level-set method. This framework utilizes: discretizations that achieve a second-order accurate level-set method on non-graded adaptive Cartesian grids for solving the associated free boundary value problem for surface diffusion; and an established library for the partitioning of the grid. We consider the cases with: irreversible aggregation, which amounts to applying Dirichlet boundary conditions at the island boundary; and an asymmetric (Ehrlich–Schwoebel) energy barrier for attachment/detachment of atoms at the island boundary, which entails the use of a Robin boundary condition. We provide the scaling analyses performed on the Stampede supercomputer and numerical examples that illustrate the capability of our methodology to efficiently simulate different aspects of epitaxial growth. The combination of adaptivity and parallelism in our approach enables simulations that are several orders of magnitude faster than those reported in the recent literature and, thus, provides a viable framework for the systematic study of mound formation on crystal surfaces.

On a viscous two-fluid channel flow including evaporation

Abstract In this contribution a particular plane steady-state channel flow including evaporation effects is investigated from analytical point of view. The channel is assumed to be horizontal. The motion of two heavy viscous immiscible fluids is governed by a free boundary value problem for a coupled system of Navier-Stokes and Stephan equations. The flow domain is unbounded in two directions and the free interface separating partially both liquids is semi-infinite, i.e. infinite in one direction. The free interface begins in some point Q where the half-line Σ 1 separating the two parts of the channel in front of Q ends. Existence and uniqueness of a suitable solution in weighted HÖLDER spaces can be proved for small data (i.e. small fluxes) of the problem.

New periodic orbits in the planar equal-mass three-body problem

It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle \begin{document}$ \mathit{\theta }$\end{document} .

Influence of process strategy on distortion and residual stresses for the powder based laser metal deposition process

AbstractLaser metal deposition (LMD) induces a complex 3‐axis residual stress state, which superimposes the external service stresses and can cause unpredicted in‐service failures. To optimize the process it is important to know the mechanisms and the opportunities to influence the occurrence of residual stresses. From the mathematical point of view, laser metal deposition represents a free boundary value problem. The track geometry is part of the solution. Previously the authors developed a thermal model that is used to calculate the time‐ and space resolved temperature distribution and the track geometry. The thermal model encompasses the powder stream, its interaction with the laser radiation, the shadowing of the laser radiation by the particles, the heating of the particles and the melt pool computation. In this publication, the evolution of residual stresses for overlapping tracks for single and multilayer processing for different powder mass rates is described.

Blowup for regular solutions and [formula omitted] solutions of Euler equations in [formula omitted] with a free boundary

The compressible Euler equations are fundamental models in the study of fluids, plasmas, condensed matter and atmospheric dynamics. In this paper, we analyze the blowup phenomena of the weakened regular solutions (ρ,u→) and the C1 solutions for γ≥2 of the Euler equations in RN in an initial bounded region Ω(0). If maxx→0∈∂Ω̄(0)∑Ni=1ui2(0,x→0)<min(2,N(γ−1))EM,where E is the total energy and M>0 is the total mass, then the corresponding solutions blow up in finite time. Our blowup development for the free boundary value problem partially complements the result for the fixed boundary problem (Makino et al. 1986).

Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options

AbstractThis paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

Moving boundary problem for the detachment in multispecies biofilms

The work presents the qualitative analysis of the free boundary value problem related to the detachment process in multispecies biofilms. In the framework of continuum approach to one-dimensional mathematical modelling of multispecies biofilm growth, we consider the system of nonlinear hyperbolic partial differential equations governing the microbial species growth, the differential equation for the biomass velocity, the differential equation that governs the free boundary evolution and also accounts for detachment, and the elliptic system for substrate dynamics. The characteristics are used to convert the original moving boundary equation into a suitable differential equation useful to solve the mathematical problem. We also provide another form of the same equation that could be used in numerical applications. Several properties of the solutions to the free boundary problem are shown, such as positiveness of the functions that describe the microbial concentrations and estimates on the characteristic functions. Uniqueness and existence of solutions are proved by introducing a suitable system of Volterra integral equations and using the fixed point theorem.

On a free boundary problem for biosorption in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the biosorption process in multispecies biofilms. In the framework of continuum biofilm modeling, the mathematical problem consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear parabolic partial differential equations describing the substrate trends and a system of semilinear parabolic partial differential equations accounting for the diffusion, reaction and biosorption of different agents on the various biofilm constituents. Two systems of nonlinear hyperbolic partial differential equations have been considered as well for modeling the dynamics of the free and bounded sorption sites. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. The main result is the existence and uniqueness of the solutions to the stated free boundary value problem, which have been derived by converting the partial differential equations to Volterra integral equations and then using the fixed point theorem. Moreover, the work is completed with numerical simulations for a real case examining the growth of a heterotrophic–autotrophic biofilm devoted to wastewater treatment and acting as a sorbing material for heavy metal biosorption.

Nonlinear integral equations for Bernoulli’s free boundary value problem in three dimensions

In this paper we present a numerical solution method for the Bernoulli free boundary value problem for the Laplace equation in three dimensions. We extend a nonlinear integral equation approach for the free boundary reconstruction (Kress, 2016) from the two-dimensional to the three-dimensional case. The idea of the method consists in reformulating Bernoulli’s problem as a system of boundary integral equations which are nonlinear with respect to the unknown shape of the free boundary and linear with respect to the boundary values. The system is linearized simultaneously with respect to both unknowns, i.e., it is solved by Newton iterations. In each iteration step the linearized system is solved numerically by a spectrally accurate method. After expressing the Fréchet derivatives as a linear combination of single- and double-layer potentials we obtain a local convergence result on the Newton iterations and illustrate the feasibility of the method by numerical examples.

Analysis of track formation during laser metal deposition

A fundamental understanding of the physical phenomena associated with the coaxial laser metal deposition (LMD) process is essential to enhance the science base and thereby improve the process itself. The interaction between the laser beam and the powder particles leads to an attenuation of the beam intensity and to a temperature rise of the powder particles before reaching the melt pool. To understand this mutual influence and their influence on the process result better, the powder gas stream was analyzed with an image-based particle density measurement system. This system illuminates the powder gas stream from the side with a line laser light source and monitors the particles plane by plane with a coaxially aligned camera through the powder nozzle. A high image rate allows to record the number and position of individual powder particles. The particle density distributions in the planes are determined by averaging over the images. With this, a statistical model can be derived with regard to the particles trajectories. Finally, with these results it is possible to analyze the shadowing of the laser beam by the powder particles and the change of the particle temperature in detail. The transmitted laser intensity distribution and mean particle temperature are used as boundary conditions for the governing equations of the mathematical model for the real LMD process. LMD represents mathematically a free boundary value problem. This means that the track geometry is part of the solution. The mathematical model is based on an integration of the time dependent heat equation and the computation of the track geometry with the nonlinear Young-Laplace equation, whereby a mass balance with regard to the particles which reach the melt pool surface is taken into account. The governing equations are solved with the Finite Element Method and the moving mesh approach is used in order to implement a track adapted meshing. The results concerning the track geometry show excellent agreement with the experimental ones which are determined by cross sections of laser material deposited tracks and images from high speed videography. The model does not use any geometrical approximation of the melt pool surface but determines the melt pool surface shape by the balance equation for capillary forces and mass balance.

Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem

We prove new variational properties of the spatial isosceles orbits in the equal-mass three-body problem and analyze their linear stabilities in both the full phase space \begin{document}$\mathbb{R}^{12}$\end{document} and a symmetric subspace Γ. We prove that each spatial isosceles orbit is an action minimizer of a two-point free boundary value problem with non-symmetric boundary settings. The spatial isosceles orbits form a one-parameter set with rotation angle θ as the parameter. This set of orbits always lies in a symmetric subspace Γ and we show that their linear stabilities in the full phase space \begin{document}$\mathbb{R}^{12}$\end{document} can be simplified to two separated sub-problems: linear stabilities in Γ and \begin{document}$(\mathbb{R}^{12} \setminus Γ) \cup \{0\}$\end{document} . By applying Roberts' symmetry reduction method, we prove that the orbits are always unstable in the full phase space \begin{document}$\mathbb{R}^{12}$\end{document} , but it is linearly stable in Γ when \begin{document}$θ ∈ [0.33π, 0.48 π] \cup [0.52 π, 0.78 π]$\end{document} .

Free boundary value problem to 3D spherically symmetric compressible Navier–Stokes–Poisson equations

In the paper, we consider the free boundary value problem to 3D spherically symmetric compressible isentropic Navier-Stokes-Poisson equations for self-gravitating gaseous stars with $\gamma$-law pressure density function for $6/5 <\gamma \leq 4/3$. For stress free boundary condition and zero flow density continuously across the free boundary, the global existence of spherically symmetric weak solutions is shown, and the regularity and long time behavior of global solution are investigated for spherically symmetric initial data with the total mass smaller than a critical mass.

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles. The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

A cut finite element method for the Bernoulli free boundary value problem

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H1 norm. Finally, we present illustrating numerical results.

Heat transfer analysis for a free boundary problem arising in n-diffusion equation

This paper presents research on a free boundary value problem arising in a nonlinear n-diffusion equation by using a homotopy analysis method (HAM). Approximate analytical solutions are obtained for special nonlinear diffusion functional coefficient (variable thermal conduction) k(s)=s^{i} for i=1, 3, 5 and convection functional coefficient h(s)=s^{j} for j=1, 4 and power law parameter of n=0.2, 0.5, 1.0, 2.5. Reliability and efficiency of the approximate solutions are verified by numerical ones showing good agreement. The effects of the power law exponent, the nonlinear diffusion or convection functional coefficients, and the free boundary parameter on the flux transport characteristics are presented graphically and analyzed in detail. The mathematical techniques employed in this paper have the significance in studying some other problems of engineering.

Well-posedness of a quasilinear evolution problem modelling MEMS with general permittivity

Subject of consideration is the analytical investigation of a coupled system of partial differential equations arising from the modelling of electrostatically actuated microelectromechanical systems with general permittivity profile. A quasilinear parabolic evolution problem for the displacement u of an elastic membrane is coupled with an elliptic free boundary value problem that determines the electrostatic potential \(\psi \) in the region between the elastic membrane and a rigid ground plate. The system is shown to be well-posed locally in time for all arbitrarily large values \(\lambda \) of the applied voltage, whereas small values of the applied voltage, which do not exceed a certain critical value \(\lambda _{*}\), do even allow globally in time existing solutions. In addition, conditions are specified which force solutions emerging from a non-positive initial deflection to stay non-positive as long as they exist.

On Trefftz’ integral equation for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of Bernoulli’s free boundary value problem for a harmonic function w in a doubly connected domain D in \(\mathbb {R}^2\) where an unknown free boundary \(\varGamma _0\) is determined by prescribed Cauchy data of w on \(\varGamma _0\) in addition to a Dirichlet condition on the known boundary \(\varGamma _1\). Our method is based on a two-by-two system of boundary integral equations for the unknown boundary \(\varGamma _0\) and the unknown normal derivative \(g=\partial _\nu w\) of w on \(\varGamma _1\). This system is nonlinear with respect to \(\varGamma _0\) and linear with respect to g and we suggest to solve it simultaneously for \(\varGamma _0\) and g by Newton iterations. We establish a local convergence result and exhibit the feasibility of the method by a few numerical examples.