Hamilton-Jacobi Partial Differential Equation Research Articles

On controlled Hamilton and Hamilton–Jacobi differential equations of higher-order

In this paper, we investigate the nonlinear dynamics associated with controlled Lagrangians involving higher-order derivatives. More precisely, we establish the controlled higher-order Hamilton ordinary differential equations (ODEs) and Hamilton–Jacobi partial differential equation (PDE) for the considered class of Lagrangians governed by higher-order derivatives of the state variables. Moreover, we formulate and prove an invariance result with respect to the state variable. In addition, in order to validate the theoretical results and to highlight their effectiveness, some illustrative applications are presented.

Research on lateral dynamics safety margins of carrier-based aircraft arresting

AbstractFor the safety problems caused by the limited landing space of the deck during the arresting process of the carrier-based aircraft, a dynamic model of the carrier-based aircraft’s landing and arresting is built. Based on the batch simulation method, the lateral dynamics safety envelope of the aircraft during the arresting was defined, and the dynamic response of the key points in the envelope during the arresting process was investigated. Subsequently, the influence of engine thrust and aircraft quality on the arresting safety envelope was studied based on reasonable safety evaluation indicators, and the safety status envelope of the deck arresting was given. Then, the particular Hamilton-Jacobi partial differential equation is used to obtain the lateral dynamics safety envelope of the carrier-based aircraft in the process of landing and arresting by backward inversion. Results indicate that engine thrust and landing quality have little effect on the yaw angle in the arresting safety boundary during the arresting. Additionally, with the engine thrust and landing quality increase, the maximum safe off-centre distance gradually decreases, and the safety boundary decreases accordingly. During the phase of landing glide, the engine thrust and quality have little effect on the maximum safe eccentric distance. When the engine thrust is increased by 40%, the maximum safe yaw angle is reduced from 0.3°, and the safety boundary is reduced by 4.2%. When the aircraftquality increases by 40%, the maximum safe yaw angle is reduced by 0.4°, and the safety boundary is reduced by 2.8%. The findings of this paper can provide framework for the research on theaircraft-to-carrier dynamic matching characteristics of the carrier-based system, and is of great significance to the research on improving the safety of the carrier-based aircraft landing arresting.

A modified level set method for topology optimization of sparsely-filled and slender structures

In structural optimization, the level set method is known as a well-established approach for shape and topology optimization. However, special care must be taken, if the design domains are sparsely-filled and slender. Using steepest descent-type level set methods, slender structure topology optimizations tend to instabilities and loss of structural cohesion. A sole step size control or a selection of more complex initial designs only help occasionally to overcome these issues and do not describe a universal solution. In this paper, instead of updating the level set function by solving a Hamilton–Jacobi partial differential equation, an adapted algorithm for the update of the level set function is utilized, which allows an efficient and stable topology optimization of slender structures. Including different adaptations, this algorithm replaces unacceptable designs by modifying both the pseudo-time step size and the Lagrange multiplier. Besides, adjustments are incorporated in the normal velocity formulation to avoid instabilities and achieve a smoother optimization convergence. Furthermore, adding filtering-like adaptation terms to the update scheme, even in case of very slender structures, the algorithm is able to perform topology optimization with an appropriate convergence speed. This procedure is applied for compliance minimization problems of slender structures. The stability of the optimization process is shown by 2D numerical examples. The solid isotropic material with penalization (SIMP) method is used as an alternative approach to validate the result quality of the presented method. Finally, the simple extension to 3D optimization problems is addressed, and a 3D optimization example is briefly discussed.

3D Reconstruction of Non-Lambertian Surfaces by Perspective Shape-From-Shading With Fast Viscosity Solution

Shape-from-shading (SFS) is an important method to reconstruct three-dimensional (3D) shape of a surface in photometry and computer vision. Lambertian surface reflectance and orthographic camera projection are two fundamental assumptions which generally result in undesirable reconstructed results since inaccurate imaging model is adopted. In this paper, we propose a new fast 3D shape reconstruction approach via the SFS method relaxing the two assumptions. To this end, the Oren-Nayar reflectance and perspective projection models are used to establish an image irradiance equation which depicts the relationship between the 3D shape of non-Lambertian surfaces and its corresponding two-dimensional (2D) shading image. Considering the light attenuation of the near point source, the image irradiance equation is transformed into a static Hamilton-Jacobi partial differential equation (PDE) by solving a quadratic equation. The viscosity solution of the resultant Hamilton-Jacobi PDE is approximated by using optimal control theory and iterative fast marching method starting from a viscosity supersolution. The performance of the proposed approach is evaluated on both synthetic and real-world images and the experimental results demonstrate that the proposed approach is accurate and fast.

Topology optimization of irregular flow domain by parametric level set method in unstructured mesh

ABSTRACT In this study, we present a novel method for the topology optimization of the irregular flow domain using a parametric level set method (PLSM). Some improvement was applied on the CS-RBFs (radial basis functions with compact support)-based PLSM to make it suitable for nonuniform mesh, expanding the range field of engineering application of the PLSM. The optimization problem is solved by a gradient-based algorithm with Stokes equations as state constraints, and the objective is set to minimize the power dissipation subject to the volume constraint of flow channels. A PLSM is introduced to avoid the direct solving of the Hamilton–Jacobi partial differential equation, which can have the potential to break through the restriction of relying on structured meshes because no finite difference scheme is required. Then, a self-adaption support radius approach is presented to allow the parametric level set to be evolved on the nonuniformed mesh, which can expand the application of the PLSM to more complicated engineering problems with irregular geometric shapes. A volume integration scheme is applied during the design sensitivity analysis to calculate the shape derivatives, allowing the nucleation of new holes. Numerical examples in two and three dimensions are provided to demonstrate the effectiveness of the proposed method.

The maximum taxiing safe set of the wheel-skid aircraft under optimal control of rudder

Directional stability during the roll-out phase is crucial to the safety and reusability of the aircraft. Because of the mechanical properties, the wheel-skid aircraft is more prone to produce course instability. To address this issue, the taxiing safe set of the wheel-skid aircraft is discussed based on the reachability theory. A dynamic model of the on-ground aircraft is established firstly, considering the complex condition of the ground loads. Then, the particular Hamilton–Jacobi partial differential equation is used to obtain the safe set. According to the safe set results, the optimal control of the rudder is built in the state space. Its effectiveness is verified by the comparison with other robust methods. In addition, three structural parameters are selected to analyze the influences on the safe set. Results indicate that the maximum safe yaw angle increases from [Formula: see text] to [Formula: see text] at 70 m/s under the optimal control of the rudder when the steering of nose wheel is locked. The safe boundary in the middle–high-speed region expands by 43.5% under the rudder control. Because of the mechanical properties, uncontrollable deflection will appear due to the asymmetric disturbances when the longitudinal velocity is lower than 42 m/s.

A Novel Unified Framework for Solving Reachability and Invariance Problems

The level set method is a widely used tool for solving reachability and invariance problems. However, some shortcomings, such as large amount of storage space consumption and the difficulty of constructing terminal conditions for solving the Hamilton-Jacobi partial differential equation, limit the application of the level set method in some problems, especially those with irregular target sets. This paper proposes a method that can effectively avoid the above tricky issues and thus has better generality. In the proposed method, the reachable or invariant sets with different time horizons are characterized by some non-zero sublevel sets of a value function. This value function is not obtained by solving a viscosity solution of the partial differential equation but by recursion and interpolation approximation. At the end of this paper, some examples are taken to illustrate the accuracy and generality of the proposed method.

Semianalytical Solutions to the Lighthill–Whitham–Richards Equation With Time-Switched Triangular Diagrams: Application to Variable Speed Limit Traffic Control

This article proposes a new approach for computing a semiexplicit form of the solution to a class of traffic flow problems encoded by a Hamilton-Jacobi (HJ) partial differential equation (PDE), with time-switched Hamiltonian. Using a characterization of the problem derived from viability theory, we show that the solution associated with the problem can be formulated as a minimization problem involving the trajectory of an auxiliary dynamical system. A generalized Lax-Hopf formula for the switched Hamiltonian problem is derived, which enables us to compute the solution associated with affine initial or boundary conditions as a linear program involving the control function of the auxiliary dynamical system. This formulation allows us to compute the solution to the original problem exactly, unlike dynamic programming methods. In addition, this method allows one to very efficiently recompute the boundary conditions associated with an initial condition problem, allowing large-scale variable speed limit traffic control problems to be solved.

Inner-Approximating Reachable Sets for Polynomial Systems With Time-Varying Uncertainties

In this paper, we propose a convex programming based method to address a long-standing problem of inner-approximating backward reachable sets of state-constrained polynomial systems subject to time-varying uncertainties. The backward reachable set is a set of states, from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties. It is equal to the zero sublevel set of the unique Lipschitz viscosity solution to a Hamilton–Jacobi partial differential equation (HJE). We show that inner approximations of the backward reachable set can be formed by zero sublevel sets of its viscosity supersolutions. Consequently, we reduce the inner-approximation problem to a problem of synthesizing polynomial viscosity supersolutions to this HJE. Such a polynomial solution in our method is synthesized by solving a single semidefinite program. We also prove that polynomial solutions to the formulated semidefinite program exist and can produce a convergent sequence of inner approximations to the interior of the backward reachable set in measure under appropriate assumptions. This is the main contribution of this paper. Several illustrative examples demonstrate the merits of our approach.

Safety Analysis for Icing Aircraft during Landing Phase Based on Reachability Analysis

Icing is one of the leading causes of fatal aircraft accidents worldwide. Encountering icing conditions, dynamic characteristics of the aircraft will be damaged, thereby greatly affecting flight safety. Research on real-time estimations of the safety envelope under icing conditions is critical to improve flight safety. In order to determine the safety envelope, the reachability analysis based on the level set method is presented. The reachable set is obtained via computing the Hamilton-Jacobi partial differential equation (HJPDE), which is based on optimal control and is used in the landing phase of an aircraft. The results show that icing will shrink the safety envelope. Particularly under severe icing conditions, the stall speed of the aircraft will increase, and dynamic behavior will be more sensitive. A slight change of command by the pilot may contribute to the flight state moving outside the safety envelope. Furthermore, the effect of flap deflection was considered, which positively impacts the expanding safety envelope during the landing phase. Finally, a maneuvering coping strategy based on safety envelope is proposed. Examples are provided using the NASA’s generic transport model (GTM), and the results can be applied to flight safety risk assessment, providing theoretical guidance for the design of envelope protection systems.

Deep relaxation: partial differential equations for optimizing deep neural networks

Entropy-SGD is a first-order optimization method which has been used successfully to train deep neural networks. This algorithm, which was motivated by statistical physics, is now interpreted as gradient descent on a modified loss function. The modified, or relaxed, loss function is the solution of a viscous Hamilton–Jacobi partial differential equation (PDE). Experimental results on modern, high-dimensional neural networks demonstrate that the algorithm converges faster than the benchmark stochastic gradient descent (SGD). Well-established PDE regularity results allow us to analyze the geometry of the relaxed energy landscape, confirming empirical evidence. Stochastic homogenization theory allows us to better understand the convergence of the algorithm. A stochastic control interpretation is used to prove that a modified algorithm converges faster than SGD in expectation.

A cortical shape-adaptive approach to local gyrification index.

The amount of cortical folding, or gyrification, is typically measured within local cortical regions covered by an equidistant geodesic or nearest neighborhood-ring kernel. However, without careful design, such a kernel can easily cover multiple sulcal and gyral regions that may not be functionally related. Furthermore, this can result in smoothing out details of cortical folding, which consequently blurs local gyrification measurements. In this paper, we propose a novel kernel shape to locally quantify cortical gyrification within sulcal and gyral regions. We adapt wavefront propagation to generate a spatially varying kernel shape that encodes cortical folding patterns: neighboring gyral crowns, sulcal fundi, and sulcal banks. For this purpose, we perform anisotropic wavefront propagation that runs fast along gyral crowns and sulcal fundi by solving a static Hamilton-Jacobi partial differential equation. The resulting kernel adaptively elongates along gyral crowns and sulcal fundi, while keeping a uniform shape over flat regions like sulcal banks. We then measure local gyrification within the proposed spatially varying kernel. The experimental results show that the proposed kernel-based gyrification measure achieves a higher reproducibility than the conventional method in a multi-scan dataset. We further apply the proposed kernel to a brain development study in the early postnatal phase from neonate to 2 years of age. In this study we find that our kernel yields both positive and negative associations of gyrification with age, whereas the conventional method only captures positive associations. In general, our method yields sharper and more detailed statistical maps that associate cortical folding with sex and gestational age.

Parametric structural shape & topology optimization with a variational distance-regularized level set method

The signed distance function (SDF) gives the shortest distance from a given point to the boundary, and the sign indicates whether this point is inside or outside the closed boundary or enclosed region. The SDF property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance-based interpolation of different material properties. In conventional level set methods, a common way of achieving a level set function with the signed distance property is to periodically implement the so-called reinitialization scheme by solving an additional Hamilton–Jacobi partial differential equation. However, such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended, which may shift the optimization result and bring convergence issues. In this paper, a double-well potential functional is employed for distance regularization inside the topology optimization loop, which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain. The radial basis function (RBF) based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance-regularized topology optimization method in handling problems with non-convex objective functions and multiple constraints. The flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process. Both 2D and 3D benchmark examples are employed to demonstrate the validity of the proposed method.

Staticization, its dynamic program and solution propagation

Stationary-action formulations of dynamical systems are considered. Use of stationary-action formulations allows one to generate fundamental solutions for classes of two-point boundary-value problems (TPBVPs). One solves for stationary points of the payoff as a function of inputs rather than minimization/maximization, a task which is significantly different from that in optimal control problems. Both a dynamic programming principle (DPP) and a Hamilton–Jacobi partial differential equation (HJ PDE) are obtained for a class of problems subsuming the stationary-action formulation. Although convexity (or concavity) of the payoff may be lost as one propagates forward, stationary points continue to exist, and one must be able to use the DPP and/or HJ PDE to solve forward to such time horizons. In linear/quadratic models, this leads to a requirement for propagation of solutions of differential Riccati equations past finite escape times. Such propagation is also required in (nonlinear) n-body problem formulations where the potential is represented via semiconvex duality. The dynamic programming tools developed here are applicable.

Revised Level Set-Based Method for Topology Optimization and Its Applications in Bridge Construction

To cure imperfections such as low accuracy and the lack of ability to nucleate hole in the conventional level set-based topology optimization method, a novel method using a trapezoidal method with discrete design variables is proposed. The proposed method can simultaneously accomplish topology and shape optimization. The finite element method is employed to obtain element properties and provide data for calculating design and topological sensitivities. With the aim of performing the finite element method on a non-conforming mesh, a relation between the level set function and the element densities field has to be clearly defined. The element densities field is obtained by averaging the Heaviside function values. The Lagrange multiplier method is exploited to fulfill the volume constraint. Based on topological and design sensitivity and the trapezoidal method, the Hamilton-Jacobi partial differential equation is updated recursively to find the optimal layout. In order to stabilize the iterations and improve the efficiency of the algorithm, re-initiation of the level set function is necessary. Then, the detailed process of a cantilever design is illustrated. To demonstrate the applications of the proposed method in bridge construction, two numerical examples of a pylon bridge design are introduced. It is shown that the results match practical designs very well, and the proposed method is a helpful tool in bridge design.

A Convex Formulation of Traffic Dynamics on Transportation Networks

This article proposes a numerical scheme for computing the evolution of vehicular traffic on a road network over a finite time horizon. The traffic dynamics on each link is modeled by the Hamilton--Jacobi (HJ) partial differential equation (PDE), which is an equivalent form of the Lighthill--Whitham--Richards PDE. The main contribution of this article is the construction of a single convex optimization program which computes the traffic flow at a junction over a finite time horizon and decouples the PDEs on connecting links. Compared to discretization schemes which require the computation of all traffic states on a time-space grid, the proposed convex optimization approach computes the boundary flows at the junction using only the initial condition on links and the boundary conditions of the network. The computed boundary flows at the junction specify the boundary condition for the HJ PDE on connecting links, which then can be separately solved using an existing semi-explicit scheme for single link HJ PDE...

Topological shape optimization design of continuum structures via an effective level set method

AbstractThis paper proposes a new level set method for topological shape optimization of continuum structure using radial basis function (RBF) and discrete wavelet transform (DWT). The boundary of the structure is implicitly represented as the zero level set of a higher-dimensional level set function. The interpolation of the implicit surface using RBF is introduced to decouple the spatial and temporal dependence of the level set function. In doing so, the Hamilton-Jacobi partial differential equation (PDE) that defines the motion of the level set function is transformed into an explicit parametric form, without requiring the direct solution of the complicated PDE using the finite difference method. Therefore, many more efficient gradient-based optimization algorithms can be applied to solve the optimization problem, via updating the expansion coefficients of the interpolant and then evolving the level set function and the boundary. Furthermore, the DWT is employed to handle the full matrix arising from t...

Topology optimization based on level set for a flexible multibody system modeled via ANCF

A topology optimization methodology is proposed for the flexible multibody system undergoing both large overall motion and large deformation. The system of concern is modeled via the absolute nodal coordinate formulation. The equivalent static load method is employed to transform the topology optimization of the nonlinear dynamic response of the system into a static one, and evaluated to adapt to the absolute nodal coordinate formulation by splitting the elastic deformations of the flexible components from the overall motions of those components. During the static topology optimization, the material interface is implicitly described as the zero level set of a higher-dimensional scalar function. Then, the semi-implicit level set method with the additive operator splitting algorithm is employed to solve the corresponding Hamilton-Jacobi partial differential equation. In addition, the expert evaluation method of weights based on the grey theory is utilized to define the objective function, and a modified augmented Lagrange multiplier method is proposed to treat the inequality volume constraint so as to avoid the oscillation and drift of the volume. Finally, two numerical examples are provided to validate the proposed methodology.

Nonlinear Local Stabilization of a Viscous Hamilton-Jacobi PDE

We consider the boundary stabilization problem of the non-uniform equilibrium profiles of a viscous Hamilton-Jacobi (HJ) Partial Differential Equation (PDE) with parabolic concave Hamiltonian. We design a nonlinear full-state feedback control law, assuming Neumann actuation, which achieves an arbitrary rate of convergence to the equilibrium. Our design is based on a feedback linearizing transformation which is locally invertible. We prove local exponential stability of the closed-loop system in the $H^{1} $ norm, by constructing a Lyapunov functional, and provide an estimate of the region of attraction. We design an observer-based output-feedback control law, by constructing a nonlinear observer, using only boundary measurements. We illustrate the results on a benchmark example computed numerically.

A level set method for topological shape optimization of 3D structures with extrusion constraints

This paper proposes a new level set method for topological shape optimization of 3D structures considering manufacturing constraints. First, the boundary of structure is implicitly represented as the zero level set of a higher-dimensional level set function, and the implicit surface is parameterized through the interpolation of a given set of compactly supported radial basis functions. In this way, the original Hamilton–Jacobi partial differential equation is transformed into a system of algebraic equations. Correspondingly, the topological shape optimization is changed to the easiest size optimization in structural optimization. Many more efficient gradient-based optimization algorithms can be directly applied to the size optimization. Second, to save the expensive computational cost in the 3D large-scale optimization problems, the discrete wavelet transform is introduced into the level set method to compress the size of the coefficient matrix in compactly supported radial basis function interpolant. The discrete wavelet transform converts the original matrix into a set of wavelet basis and therefore eliminates the “noise” elements in the new matrix, so that the linear system can be replaced by a sparser one. Finally, a cross section projection strategy is utilized to ensure the satisfaction of the extrusion constraint and reduce the number of the design variables, simultaneously. Several numerical examples in 3D structures are employed to demonstrate the effectiveness of the proposed method.