Dirichlet Energy Research Articles

A mesh regularization scheme to update internal control points for isogeometric shape design optimization

This paper presents a variational method to update internal control points in isogeometric shape optimization. The important properties of domain parameterization such as bijective mapping between parametric and physical domains, uniform mesh, and orthogonal mesh are enforced simultaneously. The bijective mapping is achieved by minimizing a Dirichlet energy functional. To prevent the divergent behavior of the minimizing process due to the severely distorted initial mesh, a constraint is introduced to enforce the positive Jacobian of mapping from parametric to physical domains. In spite of adding the constraint that might increase computational costs, the proposed method is more efficient due to the convexity of Dirichlet energy functional, compared with the other unconstrained methods. Also, it turns out that the proposed method is more effective to achieve the bijective mapping, especially near a concave boundary. The uniform parameterization of the domain is achieved by minimizing the Dirichlet energy functional and the orthogonality of mesh is performed by minimizing a dimensionless functional. The required design sensitivity of the employed functional and constraint is derived with respect to the position of internal control points. The developed scheme of mesh regularization is effective to maintain the high quality of domain parameterization during the shape optimization process.

On a functional connected to the laplacian in a family of punctured regular polygons in ℝ2

Let p1 and p0 be closed, regular, convex, concentric polygons having n sides in ℝ2 such that the circumradius of p0 is strictly less than the inradius of p1. We fix p1 and vary p0 by rotating it about its center. Let Ω be the interior of p1 p0. Let u be the solution of the stationary problem −Δu = 1 in Ω vanishing on the boundary. We show that the associated Dirichlet energy functional J(Ω) attains its extremum values when the axes of symmetry of p0 coincide with those of p1.

Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A $d$-dimensional pluri-Lagrangian problem can be described as follows: given a $d$-form $L$ on an $m$-dimensional space, $m > d$, whose coefficients depend on a function $u$ of $m$ independent variables (called field), find those fields $u$ which deliver critical points to the action functionals $S_\Sigma=\int_\Sigma L$ for any $d$-dimensional manifold $\Sigma$ in the $m$-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e. those with quadratic Lagrangians $L$. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

THE LIOUVILLE PROPERTY FOR PSEUDOHARMONIC MAPS WITH FINITE DIRICHLET ENERGY

In this paper, we first derive the CR Bochner formula and the CR Kato's inequality for pseudoharmonic maps. Secondly, by applying the CR Bochner formula and the CR Kato's inequality we are able to prove the Liouville property for pseudoharmonic maps with finite Dirichlet energy in a complete $(2n+1)$-pseudohermitian manifold. This is served as CR analogue to the Liouville theorem for harmonic maps in Riemannian Geometry.

Liouville type results for local minimizers of the micromagnetic energy

We study local minimizers of the micromagnetic energy in small ferromagnetic 3d convex particles for which we justify the Stoner–Wohlfarth approximation: given a uniformly convex shape $$\Omega \subset \varvec{\mathbf {R}}^3$$ , there exist $$\delta _c$$ >0 and $$C > 0$$ such that for $$0 < \delta \le \delta _c$$ any local minimizer $$\mathbf {m}$$ of the micromagnetic energy in the particle $$\delta \Omega $$ satisfies $$\Vert \nabla \mathbf {m} \Vert _{L^2} \leqslant C \delta ^2$$ . In the case of ellipsoidal particles we strengthen this result by proving that, for $$\delta $$ small enough, local minimizers are exactly spatially uniform. This last result extends W.F. Brown’s fundamental theorem for fine 3d ferromagnetic particles Brown (J Appl Phys 39:463–488, 1968), Di Fratta et al. (Physica B 407(9):1368–1371, 2011) which states the same result but only for global minimizers. As a by-product of the method that we use, we establish a new Liouville type result for locally minimizing $$p$$ -harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish that in a smooth uniformly convex domain of $$\mathbf {R}^d$$ any local minimizer of the $$p$$ -Dirichlet energy ( $$p > 1$$ , $$p \ne d$$ ) is constant.

Minimization of a fractional perimeter-Dirichlet integral functional

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely ∫Ω|∇u(x)|2dx+Perσ({u>0},Ω), with σ∈(0,1). We obtain regularity results for the minimizers and for their free boundaries ∂{u>0} using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.

On local well-posedness of the thin-film equation via the Wasserstein gradient flow

A local existence and uniquness of the gradient flow of one dimensional Dirichlet energy on the Wasserstein space is proved. The proofs are based on a relaxation of displacement convexity in the Wasserstein space and can be applied to a family of higher order energy functionals which are not displacement convex in the standard sense. As the result a local well-posedness of the corresponding nonlinear evolution equations including the thin-film equation and the quantum drift diffusion equation are proved.

A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts

Given a resistor network on $\mathbb Z^d$ with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.

Invertibility versus Lagrange equation for traction free energy-minimal deformations

Let \(\,{\mathbb {X}} , {\mathbb {Y}} \subset {\mathbb {R}}^2 \,\) be bounded Jordan domains of the same topological type and \(\,h : {\mathbb {X}} \xrightarrow []{{}_{\!\!\mathrm{onto\,\,}\!\!}}{\mathbb {Y}}\,\) a traction free minimal mapping for the Dirichlet energy integral. It is shown that \(\,h : {\mathcal {O}} \rightarrow {\mathbb {Y}}\,\), restricted to any subdomain \(\,{\mathcal {O}} \subset {\mathbb {X}}\,\), is injective if and only if it is harmonic in \(\,{\mathcal {O}}\,\). This result appears pertinent to other energy integrals and, in greater generality, may be interpreted as saying that the interpenetration of matter (under hyperelastic deformations of thin plates) is inevitable precisely in the localities where the Lagrange equation fails.

Some new problems in spectral optimization

We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the <i>metric Laplacian</i>, and we consider in particular Riemannian or Finsler manifol

Minimal Dirichlet Energy Partitions for Graphs

Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides a natural representative for the clusters as well.

Statistical model of stress corrosion cracking based on extended form of Dirichlet energy

Statistical model of stress corrosion cracking based on extended form of Dirichlet energy

Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Let $$M$$ and $$N$$ be doubly connected Riemann surfaces with boundaries and with nonvanishing conformal metrics $$\sigma $$ and $$\rho $$ respectively, and assume that $$\rho $$ is a smooth metric with bounded Gauss curvature $${\mathcal {K}}$$ and finite area. The paper establishes the existence of homeomorphisms between $$M$$ and $$N$$ that minimize the Dirichlet energy. Among all homeomorphisms $$f :M{\overset{{}_{ \tiny {\mathrm{onto}} }}{\longrightarrow }} N$$ between doubly connected Riemann surfaces such that $${{\mathrm{Mod\,}}}M \leqslant {{\mathrm{Mod\,}}}N$$ there exists, unique up to conformal automorphisms of M, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec et al. (Invent Math 186(3):667–707, 2011), where the authors considered bounded doubly connected domains in the complex plane w.r. to Euclidean metric.

Shape optimization problems for metric graphs

We consider the shape optimization problem $$\min\big\{{\mathcal E}(\Gamma)\ :\ \Gamma\in{\mathcal A},\ {\mathcal H}^1(\Gamma)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\mathcal D}=\{D_1,\dots,D_k\}\subset{\mathbb R}^d$. The cost functional ${\mathcal E}(\Gamma)$ is the Dirichlet energy of $\Gamma$ defined through the Sobolev functions on $\Gamma$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

Dirichlet Energy for Analysis and Synthesis of Soft Maps

AbstractSoft maps taking points on one surface to probability distributions on another are attractive for representing surface mappings in the presence of symmetry, ambiguity, and combinatorial complexity. Few techniques, however, are available to measure their continuity and other properties. To this end, we introduce a novel Dirichlet energy for soft maps generalizing the classical map Dirichlet energy, which measures distortion by computing how soft maps transport probabilistic mass from one distribution to another. We formulate the computation of the Dirichlet energy in terms of a differential equation and provide a finite elements discretization that enables all of the quantities introduced to be computed. We demonstrate the effectiveness of our framework for understanding soft maps arising from various sources. Furthermore, we suggest how these energies can be applied to generate continuous soft or point‐to‐point maps.

Mappings of Least Dirichlet Energy and their Hopf Differentials

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class $\bar{\mathscr H}_2(X, Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X, Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.

On double-covering stationary points of a constrained Dirichlet energy

The double-covering map udc:R2→R2 is given byudc(x)=12|x|(x22−x122x1x2) in cartesian coordinates. This paper examines the conjecture that udc is the global minimizer of the Dirichlet energy I(u)=∫B|∇u|2dx among all W1,2 mappings u of the unit ball B⊂R2 satisfying (i) u=udc on ∂B, and (ii) det∇u=1 almost everywhere. Let the class of such admissible maps be A. The chief innovation is to express I(u) in terms of an auxiliary functional G(u−udc), using which we show that udc is a stationary point of I in A, and that udc is a global minimizer of the Dirichlet energy among members of A whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about udc in A using ODE techniques, we also show that udc is a local minimizer among variations whose tangent ψ to A at udc obeys G(ψo)>0, where ψo is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint det∇u=1 is identified by an analysis which exploits the well-known Fefferman–Stein duality.

Convexity of Reduced Energy and Mass Angular Momentum Inequalities

In this paper, we extend the work in \cite{D}\cite{ChrusLiWe}\cite{ChrusCo}\cite{Co}. We weaken the asymptotic conditions on the second fundamental form, and we also give an $L^{6}-$norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr-Newman data by proving convexity of the renormalized Dirichlet energy when the target has non-positive curvature. In particular, we give the first proof of the strict mass/angular momentum/charge inequality for axisymmetric Einstein/Maxwell data which is not identical with the extreme Kerr-Newman solution.

GENERALIZED TWISTS AS ELASTIC ENERGY EXTREMALS ON ANNULI, QUATERNIONS AND LIFTING TWIST LOOPS TO THE SPINOR GROUPS

Let X = {x ∈ ℝn : a &lt; |x| &lt; b} be a generalized annulus and consider the Dirichlet energy functional [Formula: see text] over the space of admissible maps [Formula: see text] where φ is the identity map. In this paper we consider a class of maps referred to as generalized twists and examine them in connection with the Euler–Lagrange equation associated with 𝔽[⋅, X] on [Formula: see text]. The approach is novel and is based on lifting twist loops from SO(n) to its double cover Spin(n) and reformulating the equations accordingly. We restrict our attention to low dimensions and prove that for n = 4 the system admits infinitely many smooth solutions in the form of twists while for n = 3 this number sharply reduces to one. We discuss some qualitative features of these solutions in view of their remarkable explicit representation through the exponential map of Spin(n).

RANDOM WALKS AND KURAMOCHI BOUNDARIES OF INFINITE NETWORKS

In this paper, we study a connected non-parabolic, or transient, network compactified with the Kuramochi boundary, and show that the rand om walk converges almost surely to a random variable valued in the harmonic boundary, and a function of finite Dirichlet energy converges along the random walk to a random variable almost surely and in L 2 . We also give integral representations of solutions of Pois son equations on the Kuramochi compactification.