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On the Formalization of the Heat Conduction Problem in HOL
Abstract
Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the temperature distribution and diffusion of heat within a body. Due to the wider utility of these equations in various safety-critical applications, such as thermal protection systems, a formal analysis of the heat transfer is of utmost importance. In this paper, we propose to use higher-order-logic (HOL) theorem proving for formally analyzing the heat conduction problem in rectangular coordinates. In particular, we formally model the heat transfer as a one-dimensional heat equation for a rectangular slab using the multivariable calculus theories of the HOL Light theorem prover. This requires the formalization of the heat operator and formal verification of its various properties, such as linearity and scaling. Moreover, we use the separation of variables method for formally verifying the solution of the PDEs, which allows modeling the heat transfer in the slab under various initial and boundary conditions using HOL Light.
... We also provide the formal verification of these exact potential flow solutions for the Laplace equation, along with their applications in aerodynamics. While there exist some formalization work of other types of partial differential equations, such as the wave equation [4], the heat equation [7] and the telegrapher's equations [8], to the best of our knowledge, there is no formalization of the Laplace equation in the literature. Therefore, the formal analysis of potential flows governed by the Laplace equation using HOL theorem proving is the first of its kind, which could be very useful for safety-critical applications. ...
As the theoretical foundation of Lagrangian mechanics, Euler–Lagrange equation sets are widely applied in building mathematical models of physical systems, especially in solving dynamics problems. However, their preconditions are often not fully satisfied in practice. Therefore, it is necessary to verify their applications. The purpose of the present work is to conduct such verification by establishing a formal theorem library of Lagrangian mechanics in HOL Light. For this purpose, some basic concepts such as functional variation and the necessary conditions for functional extreme are formalized. Then, the fundamental lemma of variational calculus is formally verified and some new constuctors and destructors are proposed. Finally, the Euler–Lagrange equation set is formalized. To validate the formalization, the formalization results are applied to verify the least resistance problem of gas flow. The present work not only lays a necessary and solid foundation for application involving Lagrangian mechanics but also extends the HOL Light theorem library.
In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.
Formal analysis of ordinary differential equations (ODEs) and dynamical systems requires a solid formalization of the underlying theory. The formalization needs to be at the correct level of abstraction, in order to avoid drowning in tedious reasoning about technical details. The flow of an ODE, i.e., the solution depending on initial conditions, and a dedicated type of bounded linear functions yield suitable abstractions. The dedicated type integrates well with the type-class based analysis in Isabelle/HOL and we prove advanced properties of the flow, most notably, differentiable dependence on initial conditions via the variational equation. Moreover, we formalize the notion of first return or Poincaré map and prove its differentiability. We provide rigorous numerical algorithm to solve the variational equation and compute the Poincaré map. © 2018 Springer Science+Business Media B.V., part of Springer Nature
Control systems are an integral part of almost every engineering and physical system and thus their accurate analysis is of utmost importance. Traditionally, control systems are analyzed using paper-and-pencil proof and computer simulation methods, however, both of these methods cannot provide accurate analysis due to their inherent limitations. Model checking has been widely used to analyze control systems but the continuous nature of their environment and physical components cannot be truly captured by a state-transition system in this technique. To overcome these limitations, we propose to use higher-order-logic theorem proving for analyzing linear control systems based on a formalized theory of the Laplace transform method. For this purpose, we have formalized the foundations of linear control system analysis in higher-order logic so that a linear control system can be readily modeled and analyzed. The paper presents a new formalization of the Laplace transform and the formal verification of its properties that are frequently used in the transfer function based analysis to judge the frequency response, gain margin and phase margin, and stability of a linear control system. We also formalize the active realizations of various controllers, like Proportional-Integral-Derivative (PID), Proportional-Integral (PI), Proportional-Derivative (PD), and various active and passive compensators, like lead, lag and lag-lead. For illustration, we present a formal analysis of an unmanned free-swimming submersible vehicle using the HOL Light theorem prover.
Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations. In this paper, we present an ongoing project, which focuses on the higher-order logic formalization of transform methods using HOL Light theorem prover. In particular, we present the motivation of the formalization, which is followed by the related work. Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. Finally, we present a roadmap to achieve our objectives, the current status and the future goals for this project.
Fourier transform based techniques are widely used for solving differential equations and to perform the frequency response analysis of signals in many safety-critical systems. To perform the formal analysis of these systems, we present a formalization of Fourier transform using higher-order logic. In particular, we use the HOL-Light’s differential, integral, transcendental and topological theories of multivariable calculus to formally define Fourier transform and reason about the correctness of its classical properties, such as existence, linearity, frequency shifting, modulation, time reversal and differentiation in time-domain. In order to demonstrate the practical effectiveness of the proposed formalization, we use it to formally verify the frequency response of an automobile suspension system.
Many ordinary differential equations (ODEs) do not have a closed solution, therefore approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL.
We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf theorem. We introduce generic one-step methods for numerical approximation of the solution and provide an analysis regarding the local and global error of one-step methods.
We give an executable specification of the Euler method as an instance of one-step methods. With user-supplied proofs for bounds of the differential equation we can prove an explicit bound for the global error. We use arbitrary-precision floating-point numbers and also handle rounding errors when we truncate the numbers for efficiency reasons.
The formal verification of cyber-physical systems is a challenging task mainly because of the involvement of various factors of continuous nature, such as the analog components or the surrounding environment. Traditional verification methods, such as model checking or automated theorem proving, usually deal with these continuous aspects by using abstracted discrete models. This fact makes cyber-physical system designs error prone, which may lead to disastrous consequences given the safety and financial critical nature of their applications. Leveraging upon the high expressiveness of higher-order logic, we propose to use higher-order-logic theorem proving to analyze continuous models of cyber-physical systems. To facilitate this process, this paper presents the formalization of the solutions of second-order homogeneous linear differential equations. To illustrate the usefulness of our foundational cyber-physical system analysis formalization, we present the formal analysis of a damped harmonic oscillator and a second-order op-amp circuit using the HOL4 theorem prover.
Recent developments in autonomous driving, vehicle-to-vehicle communication and smart traffic controllers have provided a hope to realize platoon formation of vehicles. The main benefits of vehicle platooning include improved safety, enhanced highway utility, efficient fuel consumption and reduced highway accidents. One of the central components of reliable and efficient platoon formation is the underlying control strategies, e.g., constant spacing, variable spacing and dynamic headway. In this paper, we provide a generic formalization of platoon control strategies in higher-order logic. In particular, we formally verify the stability constraints of various strategies using the libraries of multivariate calculus and Laplace transform within the sound core of HOL Light proof assistant. We also illustrate the use of verified stability theorems to develop runtime monitors for each controller, which can be used to automatically detect the violation of stability constraints in a runtime execution or a logged trace of the platoon controller. Our proposed formalization has two main advantages: (1) it provides a framework to combine both static (theorem proving) and dynamic (runtime) verification approaches for platoon controllers; and (2) it is inline with the industrial standards, which explicitly recommend the use of formal methods for functional-safety, e.g., automotive ISO 26262.
Jiji's extensive understanding of how students think and learn, what they find difficult, and which elements need to be stressed is integrated in this work. He employs an organization and methodology derived from his experience and presents the material in an easy to follow form, using graphical illustrations and examples for maximum effect. The second, enlarged edition provides the reader with a thorough introduction to external turbulent flows, written by Glen Thorncraft. Additional highlights of note: Illustrative examples are used to demonstrate the application of principles and the construction of solutions. Solutions follow an orderly approach used in all examples. Systematic problem-solving methodology emphasizes logical thinking, assumptions, approximations, application of principles and verification of results. Chapter summaries help students review the material. Guidelines for solving each problem can be selectively given to students. An extensive solution manual for teachers is available on request. © 2009 Springer-Verlag Berlin Heidelberg. All rights are reserved.
Thermoelectric devices are semiconductor devices which are capable of either generating a voltage when placed in between a temperature gradient, exploiting the Seebeck effect, or producing a temperature gradient when powered by electricity, exploiting the Peltier effect. The devices are usually employed in environments with time-varying temperature differences and input/output powers. Therefore it becomes important to understand the behaviour of thermoelectric devices during thermal and electrical transients in order to properly simulate and design complex thermoelectric systems which also include power electronics and control systems.The purpose of this paper is to provide the transient solution to the one-dimensional heat conduction equation with internal heat generation that describes the transfer and generation of heat throughout a thermoelectric device. The solution proposed can be included in a model in which the Peltier effect, the thermal masses and the electrical behaviour of the system are considered too; this would be of great benefit because it would allow accurate simulations of thermoelectric systems.While the previous literature does not focus on the study of thermal transients in thermoelectric applications and usually considers constant the temperatures at the hot and cold sides, this paper proposes a dynamic exchange of heat through the hot and cold side, both in steady-state and transients. This paper also presents an analytical solution which is then computed by Matlab to simulate a physical experiment. Simulation results show excellent correlation with experimentally determined values, thus validating the solution.
It is well-known that programs may fail due to exceptional behaviors,
out-of-bound array accesses, or simply coding errors. Thus, they cannot be
blindly trusted. Scientific computing programs make no exception in that
respect, and even bring specific accuracy issues due to their massive use of
floating-point computations. Yet, it is uncommon to guarantee their
correctness. There exist methods and tools for proving the correct behavior of
programs and we have extended them for the verification of an existing
numerical analysis program. This C program implements the second-order centered
finite difference explicit scheme for solving the 1D wave equation. In fact, we
have gone much further as we have mechanically verified the convergence of the
numerical scheme in order to get a complete formal proof covering all aspects
from partial differential equations to actual numerical results. To the best of
our knowledge, this is the first time such a comprehensive proof is achieved.
The transient response of one-dimensional multilayered composite conducting slabs to sudden variations of the temperature of the surrounding fluid is analysed. The solution is obtained applying the method of separation of variables to the heat conduction partial differential equation. In separating the variables, the thermal diffusivity is retained on the side of the modified heat conduction equation where the time-dependent function is collected. This choice is the essence of composite medium analysis itself. In fact, it ‘naturally’ gives the relationship between the eigenvalues for the different regions and then yields a transcendental equation for the determination of the eigenvalues in a less complex form than the ones resulting from the application of traditional techniques. A new type of orthogonality relationship is developed by the author and used to obtain the final complete series solution. The errors, which develop when the higher terms in the series solution are neglected, are also investigated. Some calculated results of a numerical example are shown in a graphical form, by using dimensionless groups, and therefore discussed.
The sheer complexity of computer systems has meant that automated reasoning, i.e. the ability of computers to perform logical inference, has become a vital component of program construction and of programming language design. This book meets the demand for a self-contained and broad-based account of the concepts, the machinery and the use of automated reasoning. The mathematical logic foundations are described in conjunction with practical application, all with the minimum of prerequisites. The approach is constructive, concrete and algorithmic: a key feature is that methods are described with reference to actual implementations (for which code is supplied) that readers can use, modify and experiment with. This book is ideally suited for those seeking a one-stop source for the general area of automated reasoning. It can be used as a reference, or as a place to learn the fundamentals, either in conjunction with advanced courses or for self study.
Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent. We present a comprehensive formalization of the simplest scheme and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical pen-and-paper proofs. To our knowledge, this is the first time this kind of mathematical proof is machine-checked. Key words: partial differential equation, acoustic wave equation, numerical scheme, Coq formal proofs 1
Analytical solution for transient thermal response of an insulated structure
- M L Blosser
Handbook of Numerical Heat Transfer
- W Minkowycz
- E M Sparrow
- G E Schneider
- R H Pletcher
Applied Engineering Analysis
- T R Hsu