Nonlinear Heat Equation Research Articles

Observer design for a nonlinear heat equation: Application to semiconductor wafer processing

In this paper, the problem of observer design for a class of 1D nonlinear heat equations with pointwise in-domain temperature measurements is addressed. A pointwise measurement injection observer is designed and the robust convergence of its estimation error in presence of bounded distributed perturbations is established by verifying input-to-state stability. The obtained convergence conditions express the underlying interplay between heat conduction and radiation and include specific dependencies on the sensor locations which are the main degrees of freedom in the design approach. The theoretical results are experimentally validated on a semiconductor wafer processing unit.

Space-time hp-finite elements for heat evolution in laser powder bed fusion additive manufacturing

The direct numerical simulation of metal additive manufacturing processes such as laser powder bed fusion is challenging due to the vast differences in spatial and temporal scales. Classical approaches based on locally refined finite elements combined with time-stepping schemes can only address the spatial multi-scale nature and provide only limited scaling potential for massively parallel computations. We address these shortcomings in a space-time Galerkin framework where the finite element interpolation also includes the temporal dimension. In this setting, we construct four-dimensional meshes that are locally refined towards the laser spot and allow for varying temporal accuracy depending on the position in space. By splitting the mesh into conforming time-slabs, we recover a stepwise solution to solve the space-time problem locally in time at this slab; additionally, we can choose time-slab sizes significantly larger than classical time-stepping schemes. As a result, we believe this setting to be well suited for large-scale parallelization. In our work, we use a continuous Galerkin–Petrov formulation of the nonlinear heat equation with an apparent heat capacity model to account for the phase change. We validate our approach by computing the AMB2018-02 benchmark, where we obtain an excellent agreement with the measured melt pool shape. Using the same setup, we demonstrate the performance potential of our approach by hatching a square area with a laser path length of about one meter.

A solution to the simplified multi-dimensional energy-transport model with a general conductivity for semiconductors

The Dirichlet–Neumann mixed boundary value problem of a simplified multi-dimensional energy-transport model for semiconductors with the conductivity κ(n,θ)=nθ is studied. It consists of a drift–diffusion equation for carrier’s density, involving temperature gradients, a nonlinear heat equation for the carrier’s temperature, and the Poisson equation for the electric potential. The existence of weak solution and zero energy relaxation time limit for the problem are proved. For a small variation of the lattice temperature, the resulting limit for the problem is well-known drift–diffusion model. The proofs are based on the analysis of a time-discrete approximate system for the transient problem with the Stampacchia’s truncation approach and some careful calculations concerning the stability estimates needed for convergence of the scheme. Under some regularity assumption of the solution, the uniqueness of solution is shown.

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows

We demonstrate that the reformulation of renormalization group (RG) flow equations as nonlinear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional ${\mathbb{Z}}_{2}$-symmetric model that the dissipative character of generic nonlinear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semigroup property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dissipative character of RG flows, its irreversibility and the entropy production during the RG flow may be linked to the existence of a so-called $\mathcal{C}\text{\ensuremath{-}}/\mathcal{A}$-function. In total, this introduces an asymmetry in the so-called RG time---in complete analogy to the thermodynamic arrow of time---and allows for an interpretation of infrared actions as equilibrium solutions of dissipative RG flows equations. The impossibility of resolving microphysics from macrophysics is evident in this framework. Furthermore, we directly link the irreversibility and the entropy production in RG flows to an explicit numerical entropy production, which is manifest in diffusive and non-linear partial differential equations (PDEs) and a standard mathematical tool for the analysis of PDEs. Using exactly solvable zero-dimensional ${\mathbb{Z}}_{2}$-symmetric models, we explicitly compute the (numerical) entropy production related to the total variation nonincreasing property of the PDE during RG flows toward the infrared limit. Finally, we discuss generalizations of our findings and relations to the $\mathcal{C}\text{\ensuremath{-}}/\mathcal{A}$-theorem as well as how our work may help to construct truncations of RG flow equations in the future, including numerically stable schemes for solving the corresponding PDEs.

Geometric nonlinear transient analysis of mechanically and thermally shocked functionally graded shell panels

The present paper investigates the von-Karman geometrically nonlinear structural responses of ceramic-metal functionally graded shell panels under the effect of the cases of mechanical and thermal shocks on their ceramic-rich surface. The constituents in the FGM are temperature-dependent and are graded as per the recently proposed four-parameter power laws while their effective properties in the shells are obtained using the Mori-Tanaka homogenization scheme. The transient thermoelastic problem is solved by importing the temporally varying temperature distribution from the nonlinear 1D heat equation as thermal loads into the FE code developed for thin and moderately thick shells based on higher-order shear deformation theory. The nonlinearity in the heat equation is linearized using Picard method in each Crank-Nicholson time-step while the linearization step in the geometric nonlinear problem uses Newton-Raphson method in each Newmark time-step. The new results presented in the manuscript study the effect of geometric nonlinearity on the responses of FG shell panels and the effect of different parameters of four-parameter FG laws. It was observed that including geometric nonlinearity significantly affects the responses of the shell panel structures when the loading direction, either mechanical or thermal, along with the gradation direction is reversed.

Non-delay limit in the energy space from the nonlinear damped wave equation to the nonlinear heat equation

We consider a singular limit problem from the damped wave equation with a power type nonlinearity (NLDW) to the corresponding heat equation (NLH). We call our singular limit problem non-delay limit. We show that the solution of NLDW goes to the one of NLH in [Formula: see text] topology under the both [Formula: see text] regularity solutions. We also obtain the positive convergence rate in the weaker topology [Formula: see text]. Moreover, with restriction of the range of power, if the solution to NLH is global and decays to zero, then we get the global-in-time uniform convergence of the non-delay limit.

Generalized Space-Time Fractional Stochastic Kinetic Equation

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in Rd with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving fractional derivatives in time and fractional Laplacian in space. We firstly give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. Furthermore, we also study various properties of the solution, such as Hölder regularity, the upper bound of second moment, and the stationarity with respect to the spatial variable in the case of linear additive noise.

Wellposedness and stability of fractional stochastic nonlinear heat equation in Hilbert space

In this work, we interest in the study of the wellposedness and the stability of fractional stochastic nonlinear heat equation in the Hilbert space $$ L^{2}(0,1) $$ ; perturbed by a trace-class noise and driven by the fractional Laplacian. Precisely, we use the fixed point theorem to prove the wellposedness of the problem. Moreover, we prove the $$ p^{th} $$ -moment exponential stability and the almost surely exponential stability by imposing an additional assumption. Some examples are considered in order to confirm and support the validity of our theoretical results.

Exact Solutions with Generalized Separation of Variables in the Nonlinear Heat Equation

Exact Solutions with Generalized Separation of Variables in the Nonlinear Heat Equation

The Flow of Polynomial Roots Under Differentiation

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian $$(-\Delta )^{1/2}$$ .

Barycentric Rational Collocation Method for Nonlinear Heat Conduction Equation

Nonlinear heat equation solved by the barycentric rational collocation method (BRCM) is presented. Direct linearization method and Newton linearization method are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of nonlinear heat conduction equation is also obtained. Several numerical examples are provided to valid our schemes.

Refined blow-up asymptotics for a perturbed nonlinear heat equation with a gradient and a non-local term

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1,2], we constructed a solution u for that equation such that u and ∇u both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.

On nonlinear Miyadera-Voigt perturbations

Let \(A,C,P:D(A)\subset X\rightarrow X\) be linear operators on a Banach space X such that \(-A\) generates a strongly continuous semigroup on X, and \(F:X\rightarrow X\) be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form \({\dot{u}}(t)=G(u(t))\), where \(G:D(A)\rightarrow X\) is a nonlinear map defined by \(G=-A+C+F\circ P\). In fact, using the concept of maximal \(L^p\)-regularity and a fixed point theorem, we establish the existence and uniqueness of a strong solution for the above-mentioned semilinear equation. We illustrate our results by applications to nonlinear heat equations with respect to Dirichlet and Neumann boundary conditions, and a nonlocal unbounded nonlinear perturbation.

Rigorous numerics for nonlinear heat equations in the complex plane of time

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

A forward-backward SDE from the 2D nonlinear stochastic heat equation

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\varepsilon>0$ but divided by a factor of $\sqrt{\log\varepsilon^{-1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the "weak noise" regime. We show that, as $\varepsilon\downarrow0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras (Ann. Appl. Probab. 27(5):3050--3112, 2017).

A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

Compressible high-pressure lubrication flows in thrust bearings

We present a detailed derivation of the Reynolds equation and its corresponding energy equation for three-dimensional, steady, laminar, compressible flows of single-phase Navier–Stokes fluids in thrust bearings. These equations are shown to be valid over most of the dense and supercritical gas regime except for the vicinity of the thermodynamic critical point. It is shown that the primary thermodynamic function governing the lubrication flow of high-pressure gases is the effective bulk modulus defined as the ratio of the bulk modulus to the shear viscosity. Numerical solutions to our Reynolds equation are obtained using a finite difference scheme for both moderate and high-speed flows. Approximate solutions to our Reynolds equation for high-speed flows are also derived through a perturbation analysis. It is found that boundary layers form on three out of four edges of the thrust pad. At the inner and outer radii of the pad, the flow is governed by a nonlinear heat equation. As the main flow leaves the pad, the flow is governed by a nonlinear relaxation equation. These three boundary layer solutions are rendered consistent by the construction of boundary layer solutions in the corner regions. A composite solution is developed which provides a single approximation and has the same accuracy as the individual approximations in their respective regions of validity.

Local times for systems of non-linear stochastic heat equations

We consider \(u(t,x)=(u_1(t,x),\cdots ,u_d(t,x))\) the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by a d-dimensional space-time white noise. We prove that, when \(d\le 3\), the local time \(L(\xi ,t)\) of \(\{u(t,x)\,,\;t\in [0,T]\}\) exists and \(L(\cdot ,t) \) belongs a.s. to the Sobolev space \( H^{\alpha }({\mathbb {R}}^d)\) for \(\alpha <\frac{4-d}{2}\), and when \(d\ge 4\), the local time does not exist. We also show joint continuity and establish Hölder conditions for the local time of \(\{u(t,x)\,,\;t\in [0,T]\}\). These results are then used to investigate the irregularity of the coordinate functions of \(\{u(t,x)\,,\;t\in [0,T]\}\). Comparing to similar results obtained for the linear stochastic heat equation (i.e., the solution is Gaussian), we believe that our results are sharp. Finally, we get a sharp estimate for the partial derivatives of the joint density of \((u(t_1,x)-u(t_0,x),\cdots ,u(t_n,x)-u(t_{n-1},x))\), which is a new result and of independent interest.

Limit theorems for time-dependent averages of nonlinear stochastic heat equations

We study limit theorems for time-dependent averages of the form Xt:=12L(t)∫−L(t)L(t)u(t,x)dx, as t→∞, where L(t)=exp(λt) and u(t,x) is the solution to a stochastic heat equation on ℝ +×ℝ driven by space-time white noise with u0(x)=1 for all x∈ℝ. We show that for Xt the weak law of large numbers holds when λ>λ1, the strong law of large numbers holds when λ>λ2, the central limit theorem holds when λ>λ3, but fails when λ<λ4≤λ3, the quantitative central limit theorem holds when λ>λ5, where λi’s are positive constants depending on the moment Lyapunov exponents of u(t,x).

Reduction Method and New Exact Solutions of the Multidimensional Nonlinear Heat Equation

Reduction Method and New Exact Solutions of the Multidimensional Nonlinear Heat Equation