Search

Based on the idea of a surface moving frame in differential geometry, a surface envelopment approximation method is proposed for the integrated design of point-contact tooth surfaces. This method utilizes the envelopment characteristic curve of the first tooth surface as the spline curve and adopts the local structure of the second tooth surface along a predesigned contact path as the surface interpolation condition. Through motion transformation described by the motion invariants of the first tooth surface, a conjugate motion space for the second tooth surface is fully defined by the motion invariants of the first tooth surface. This constitutes the basis of the integrated optimization design space and ensures the global optimization and machinability of the tooth surface design method. Using the experimental data of the point meshing tooth surface loading contact, the gap between the two tooth surfaces during no-load meshing is used as the design target parameter to predict and control the shape and size of the contact area under heavy load and further the symmetry requirements of the tooth surface design. Consequently, a variational inequality model for the global optimal design of the point meshing tooth surface is established. [ABSTRACT FROM AUTHOR]

The problem of machining complex surfaces with non-ball-end cutters by strip-width-maximization machining is formulated as a kind of surface fitting problem in which the tool surface envelope feature line approximates the design surface under the movement transform. The theory of surface envelope−approximation is proposed as a general method for optimizing tool movement in single-contact strip-width-maximization machining of sculptured surfaces with non-ball-end cutters. Based on the surface moving frame, the velocity equations and transformation matrices for the tool motion relative to the workpiece, described by the motion-invariant parameters of the tool surface and design surface, are derived. A functional extremum model for optimizing the tool position ensures continuous and symmetrical motion relative to the workpiece to achieve the highest machining efficiency and accuracy. Finally, a Matlab-based simulation example verifies the machining efficiency and accuracy of the envelope approximation theory. [ABSTRACT FROM AUTHOR]

In this article, three new orthogonal frames are defined for the curves lying on a surface. These moving frames, obtained based on the Darboux frame, are called “Osculator Darboux Frame”, “Normal Darboux Frame” and “Rectifying Darboux Frame”, respectively. Also, the Osculator Darboux Frame components and curvatures are calculated for a presented example. [ABSTRACT FROM AUTHOR]

An accepted approach to computing laser-induced peak surface temperature is to employ the enthalpy formulation of the transient heat conduction equation [Grigoropoulos et al., Adv. Heat Transfer 28, 75–144 (1996); Sawyer et al., J. Laser Appl. 29, 022212 (2017)]. This approach is generally implemented using an explicit numerical scheme to solve the thermal transport equation. While it offers the advantage of modeling the solid-melt phase transition automatically, the approach results in instability-like behavior in the computed surface temperature. When laser-induced ablation becomes significant, the heating rate in the surface cell becomes unrealistically large. This results in spikes in the computed peak surface temperature due to large errors in calculating the heating rate. In this paper, we present a new approach, which we refer to as the Moving Frame Solver, that employs a moving-coordinate frame of reference, located at the receding evaporating surface. We also use an analytical representation for the phase transition region of the enthalpy-temperature relationship. The Moving Frame Solver combined with an implicit scheme leads to a stable solution without surface temperature, pressure, or velocity spikes. In other words, any instability in these computed parameters due to use of an explicit scheme (such as Dufort–Frankel) has been eliminated. Details of the new thermal solver and example calculations are presented. Numerical experiments suggest that the surface cell size needs to be small, ∼0.1 μm, to obtain a highly accurate solution with a typical metal such as aluminum. Using the Moving Frame Solver with a refined grid near the surface, but coarse elsewhere, enables accurate and stable surface temperature computation. [ABSTRACT FROM AUTHOR]

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications. [ABSTRACT FROM AUTHOR]

This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map point in directions that are most pertinent to the infinitesimal deformation of the underlying function and that the singular values measure the rate of deformation in the corresponding directions. A continuous adaption of these singular vectors, therefore, constitutes a natural moving frame that carries indwelling information of the variation. This structure exists in any dimensional space, but the development of the fundamental theory and algorithm for surface exploration is an important first step for immediate application and further generalization. In this case, trajectories of these singular vectors, referred to as singular curves, unveil some intriguing patterns per the given function. At points where singular values coalesce, curious and complex behaviors occur, manifesting specific landmarks for the function. Upon analyzing the dynamics, it is discovered that there is a remarkably simple and universal structure underneath all smooth two-parameter maps. This work delineates graphs with this interesting dynamical system and the possible new discovery that, analogous to the double helix with two base parings in DNA, two strands of critical curves and eight base pairings could encode properties of a generic and arbitrary surface. This innate structure suggests that this approach could provide a unifying paradigm in functional genetics, where all smooth surfaces could be genome-sequenced and classified accordingly. Such a concept has sparked curiosity and warrants further investigation. [ABSTRACT FROM AUTHOR]

In the present paper, we compute the conservation laws of the Vaidya-Bonner geodesic space-time metric in aRiemannian space and carry out the moving frame method for this metric. We obtain the connection forms andcurvature 2-forms, using the first and second Cartan's structure equations. Finally, the Ricci scalar tensor and the components of Einstein curvature are calculated.

In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the one-dimensional nonlinear Klein-Gordon equation. Noether's First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the Klein-Gordon equation, we first present a Lagrangian whose Euler-Lagrange equation is the Klein-Gordon equation, and then according to Goncalves and Mansfield's method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action. [ABSTRACT FROM AUTHOR]

We present a method to locally reconstruct dense video depth maps of a non-rigidly deformable object directly from a video sequence acquired by a static orthographic camera. The estimation of depth is performed locally on spatiotemporal patches of the video, and then, the full depth video is recovered by combining them together. Since the geometric complexity of a local spatiotemporal patch of a deforming non-rigid object is often simple enough to be faithfully represented with a parametric model, we artificially generate a database of small deforming rectangular meshes rendered with different material properties and light conditions, along with their corresponding depth videos, and use such data to train a convolutional neural network. Since the database images are rendered with an orthographic camera model, linear deformations along the optical axis cannot be recovered from the training images. These are known in the literature as generalized bas-relief (GBR) transformations. We address this ambiguity problem by employing the invariant-theoretic normalization procedure in order to obtain complete invariants with respect to this group of transformations, and use them in the loss function of a neural network. We tested our method on both synthetic and Kinect data and experimentally observed that the reconstruction error is significantly lower than the one obtained using conventional non-rigid structure from motion approaches and state-of-the-art video depth estimation techniques. [ABSTRACT FROM AUTHOR]

In this paper, we define three different rotation-minimizing frames by rotating the moving frame around its coordinate axis vectors. Darboux vectors associated with these frames are obtained as special cases of the Darboux vector associated with the moving frame. Using these Darboux vectors and the moving frame we define six different developable surfaces. For each of these surfaces we give two invariants of curves on these surfaces to characterize their singularities. Moreover, we show that the base curves of these surfaces are contour generators with respect to an orthogonal projection or a central projection if and only if one of the invariants given for each surface is constantly equal to zero. Examples are provided to illustrate our theorems and results. [ABSTRACT FROM AUTHOR]

The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented by \"Ozen et al. (JTAM 57(2)(2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future.

For a given subspace as a solution space of a linear ODE, we define a special linear parametric group action and prolong it to the jet bundle. We determine these group parameters by moving frame method and prove that these group parameters are the first integrals of the given ODE. These first integrals are used to construct the general form of operators which preserve given subspace invariant. [ABSTRACT FROM AUTHOR]

The cell's movement and morphological change are two interrelated cellular processes. An integrated analysis is needed to explore the relationship between them. However, it has been challenging to investigate them as a whole. The cell's trajectory can be described by its speed, curvature, and torsion. On the other hand, the three-dimensional (3D) cell shape can be studied by using a shape descriptor such as spherical harmonic (SH) descriptor, which is an extension of a Fourier transform in 3D space. We propose a novel method using parallel-transport (PT) to integrate these shape-movement data by using moving frames as the 3D-shape coordinate system. This moving frame is purely determined by the velocity vector. On this moving frame, the movement change will influence the coordinate system for shape analysis. By analyzing the change of the SH coefficients over time in the moving frame, we can observe the relationship between shape and movement. We illustrate the application of our approach using simulated and real datasets in this paper.

Kirchhoff type shells are continuum models used to study the mechanics of thin elastic bodies; these are largely based on the theory of surfaces. Here, we report a reformulation of Kirchhoff shells using the theory of moving frames. This reformulation permits us to treat the deformation and the geometry of the shell as two separate entities. The structure equations which represent the familiar torsion and curvature free conditions (of the ambient space) are used to combine deformation and geometry in a compatible way. From such a perspective, Kirchhoff type theories have non-classical features which are similar to the equations of defect mechanics (theory of dislocations and disclinations). Using the proposed framework, we solve a boundary value problem and thus demonstrate, to an extent, the importance of moving frames. [ABSTRACT FROM AUTHOR]

In this paper, a novel denoising approach based on optimal trilateral filtering using Grey Wolf Optimization (GWO) is proposed. At first, a database of noisy images are generated by adding Gaussian noise, Salt & Pepper noise and Random noise to the captured image. The filtering of noisy images are performed by Block-matching and 3D filtering (BM3D) algorithm over the components of image obtained through the moving frame approach. Then, using optimal trilateral filtering, the denoised images are reconstructed. Therefore, by using a two-level filtering approach such as Moving frame-based Block-matching and 3D filtering (BM3D) and Optimal trilateral filtering the noisy images are decomposed. The proposed optimal trilateral filter employs Grey Wolf Optimization algorithm for selecting the parameters optimally to improve the efficiency of filtering method which also reduces the time required for manual computation. The performance of the proposed image denoising algorithm is analyzed using multiple datasets and the analysis of results were done in contrast with existing conventional approaches. The results validated that the optimal trilateral filtering approach outperforms other conventional methods in terms of Mean-Square Error (MSE) and the Peak Signal-to-Noise Ratio (PSNR). [ABSTRACT FROM AUTHOR]

In this paper, by the affine analogue of the fundamental theorem for Euclidean planar curves, we classify the affine curves with constant affine curvatures. Note that we use the fully affine group and not the equi-affine subgroup consisting of area-preserving affine transformations. (Caution: much of the literature omits the "equi-" in their treatment.) According to the equivariant method of moving frames, explicit formulas for the generating affine differential invariants and invariant differential operators are constructed. At the same time, by using the fact that the affine transformation group GA (2 , R) can factor as a product of two subgroup B · SE (2 , R) and the moving frame of the subgroup SE (2 , R) , we build the moving frame of GA (2 , R) and obtain the relations among invariants of group GA (2 , R) and its subgroup SE (2 , R) . Applying the affine curvature to recognize affine equivalent objects is considered in the last part of this paper. [ABSTRACT FROM AUTHOR]

A smooth surface is considered which has a curved boundary. A system of exterior differential forms is introduced which describes the surface and boundary curves completely in the moving frame approach. A total free energy functional is defined based on these forms for which an equilibrium equation and boundary conditions of the surface are derived by calculating the variation of the total free energy. These results can be applied to a surface with several freely exposed edges. [ABSTRACT FROM AUTHOR]

Starting with triply orthogonal moving frames in 3-dimensional Lie algebras, we build Weierstrass representations for triply orthogonal and conformal coordinate systems in Euclidean and Lorentian 3-space. Our constructions include spacelike and timelike Lorentzian systems. We express the first and second fundamental forms and the mean and Gaussian curvatures for the three families of surfaces in each system in terms of the Weierstrass data. Specializing to conformal mappings, we determine the Weierstrass data for all triply conformal transformations, classically known as the generalized Möbius transformations. [ABSTRACT FROM AUTHOR]

The two-dimensional surface V2 is considered in the four-dimensional extended tffine sptce A4 = A4 U A3*, where A3* is a three-dimensional improper hyperplane carrying the structure of the projective space P3. A moving frame R = ( A, Ai, Aα)(i = 1,2; α = 3,4) is attached to the surface V2 according to a certain rule, where A ∈ V2, {Ai, Aα} ⊂ A3*. We choose a conjugated net (ω¹, ω²) arranged by geodesic lines on the surface V2 so that the straight lines (AAi) represent tangent to the lines ωi. When the point A moves on the surface V2 then each of the points Ai, Aα describes two-dimensional surfaces (Ai), (Aα) in the hyperplane A3*, on which naturally arise nets of lines (ω¹, ω²). The cases when the nets are asymptotic are considered. [ABSTRACT FROM AUTHOR]

A variational approach is given which can be applied to functionals of a general form to determine a corresponding Euler–Lagrange or shape equation. It is the intention to formulate the theory in detail based on a moving frame approach. It is then applied to a functional of a general form which depends on both the mean and Gaussian curvatures as well as the area and volume elements of the manifold. Only the case of a two-dimensional closed manifold is considered. The first variation of the functional is calculated in terms of the variations of the basic variables of the manifold. The results of the first variation allow for the second variation of the functional to be evaluated. [ABSTRACT FROM AUTHOR]

Ｍｏｓｔ ａｌｇｏｒｉｔｈｍｓ ｆｏｒ ｄｅｓｃｒｉｂｉｎｇ ｔｈｒｅｅ-ｄｉｍｅｎｓｉｏｎａｌ Ｈｉｌｂｅｒｔ ｃｕｒｖｅｓ ｃａｌｃｕｌａｔｅ ｎｏｄｅ ｃｏｏｒｄｉｎａｔｅｓ ｆｒｏｍ ｔｈｅ ｓｔａｒｔ ｐｏｉｎｔ ｔｏ ｔｈｅ ｅｎｄ ｐｏｉｎｔ ｒｅｃｕｒｓｉｖｅｌｙ. Ｄｉｒｅｃｔｉｎｇ ａｔ ｔｈｅ ｍｕｌｔｉｐｌｅ ｉｔｅｒａｔｉｏｎꎬ ａ ｎｅｗ ａｌｇｏｒｉｔｈｍ ｗａｓ ｂｒｏｕｇｈｔ ｆｏｒｔｈ. Ｂｙ ｍｅａｎｓ ｏｆ ｃｏｎｓｔｒｕｃｔｉｎｇ ｍｏｖｉｎｇ ｆｒａｍｅꎬ ｔｈｅ ｉｎｖａｒｉａｎｔｓ ｕｎｄｅｒ ｒｉｇｉｄ ｂｏｄｙ ｍｏｔｉｏｎ ａｒｅ ｏｂｔａｉｎｅｄꎬ ｔｈａｔ ｉｓꎬ ｄｉｓｃｒｅｔｅ ｃｕｒｖａｔｕｒｅ ａｎｄ ｔｏｒｓｉｏｎ. Ｃｏｎｓｉｄｅｒｉｎｇ ｍｏｖｉｎｇ ｆｒａｍｅꎬ ｔｈｅ ｎｏｄｅｓ ａｒｅ ｒｅｃｏｄｅｄ. Ｅｓｔａｂｌｉｓｈｉｎｇ ａ ｍａｐ ｂｅｔｗｅｅｎ ｔｈｅ ｉｎｆｌｅｃｔｉｏｎ ｐｏｉｎｔ ｌｏｃａｔｉｏｎ ｎｕｍｂｅｒ ａｎｄ ｔｈｅ ｄｉｓｃｒｅｔｅ ｃｕｒｖａｔｕｒｅ ａｎｄ ｔｏｒｓｉｏｎ ｏｆ ｔｈｅ ｉｎｆｌｅｃｔｉｏｎ ｐｏｉｎｔꎬ ｂａｓｅｄ ｏｎ ｔｈａｔꎬ ｗｒｉｔｉｎｇ ｔｈｅ ｃｏｒｒｅｓｐｏｎｄｉｎｇ ａｌｇｏｒｉｔｈｍ ｔｏ ｍａｋｅ ｉｔ ｆｏｒ ａｎｙ ｎｕｍｂｅｒ ｎꎬ ｔｈｅ ｐａｉｒｓ ｏｆ ｃｕｒｖａｔｕｒｅ ｔｏｒｓｉｏｎ ａｎｄ ｔｈｅ ｉｍａｇｅ ｓｔｒｕｃｔｕｒｅ ｃｏｒｒｅｓｐｏｎｄｉｎｇ ｔｏ ｔｈｅ ｉｎｆｌｅｃｔｉｏｎ ｐｏｉｎｔｓ ｃａｎ ｂｅ ｏｕｔｐｕｔ. Ｃｏｍｐａｒｅｄ ｔｏ ｔｈｅ ａｌｇｏｒｉｔｈｍ Ｈｉｌｂｅｒｔ３ (ｎ)ꎬ ｔｈｅ ｐｒｏｐｏｓｅｄ ａｌｇｏｒｉｔｈｍ ｉｓ ｎｏｔ ｌｉｍｉｔｅｄ ｔｏ ｔｈｅ ｏｒｄｅｒ ｏｆ ｔｈｅ ｃｕｒｖｅ ａｎｄ ｄｏｅｓ ｎｏｔ ｄｅｐｅｎｄ ｏｎ ｔｈｅ ｉｔｅｒａｔｉｏｎ ｂｅｔｗｅｅｎ ｔｈｅ ｃｏｏｒｄｉｎａｔｅｓ. Ｅｘｐｅｒｉｍｅｎｔａｌ ｒｅｓｕｌｔｓ ｓｈｏｗ ｔｈａｔ ｔｈｅ ａｌｇｏｒｉｔｈｍ ｉｓ ｍｏｒｅ ｅｆｆｉｃｉｅｎｔ. [ABSTRACT FROM AUTHOR]

We use moving frames to construct and classify the joint invariants and joint differential invariants of binary and ternary forms. In particular, we prove that the differential invariant algebra of ternary forms is generated by a single third order differential invariant. To connect our results with earlier analysis of Kogan, we develop a general method for relating differential invariants associated with different choices of cross-section. [ABSTRACT FROM AUTHOR]

The geometry and topology of the region in which a director field is embedded impose limitations on the kind of supported orientational order. These limitations manifest as compatibility conditions that relate the quantities describing the director field to the geometry of the embedding space. For example, in two dimensions the splay and bend fields suffice to determine a director uniquely (up to rigid motions) and must comply with one relation linear in the Gaussian curvature of the embedding manifold. In 3D there are additional local fields describing the director, i.e. fields available to a local observer residing within the material, and a number of distinct ways to yield geometric frustration. So far it was unknown how many such local fields are required to uniquely describe a 3D director field, nor what are the compatibility relations they must satisfy. In this work, we address these questions directly. We employ the method of moving frames to show that a director field is fully determined by five local fields. These fields are shown to be related to each other and to the curvature of the embedding space through six differential relations. As an application of our method, we characterize all uniform distortion director fields, i.e., directors for which all the local characterizing fields are constant in space, in manifolds of constant curvature. The classification of such phases has been recently provided for directors in Euclidean space, where the textures correspond to foliations of space by parallel congruent helices. For non-vanishing curvature, we show that the pure twist phase is the only solution in positively curved space, while in the hyperbolic space uniform distortion fields correspond to foliations of space by (non-necessarily parallel) congruent helices. Further analysis of the obtained compatibility fields is expected to allow to also construct new non-uniform director fields.

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a "radius" function and a "pitch" function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of affine curvatures on a discrete planar curve. By the moving frame method, they are in fact the discrete Maurer–Cartan invariants. It shows that two curves with the same curvature sequences are affinely equivalent. Conditions for the curves with some obvious geometric properties are obtained and examples with constant curvatures are considered. On the other hand, by using the affine invariants and optimization methods, it becomes possible to collect the IFSs of some self-affine fractals with desired geometrical or topological properties inside a fixed area. In order to estimate their Hausdorff dimensions, GPUs can be used to accelerate parallel computing tasks. Furthermore, the method could be used to a much broader class. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to use the theory of exterior differential forms in deriving variations of the deformed Willmore energy in space forms and study the minimizers of the deformed Willmore energy in space forms. We derive both first and second order variations of deformed Willmore energy in space forms explicitly using moving frame method. We prove that the second order variation of deformed Willmore energy depends on the intrinsic Laplace Beltrami operator, the sectional curvature and some special operators along with mean and Gauss curvatures of the surface embedded in space forms, while the first order variation depends on the extrinsic Laplace Beltrami operator. [ABSTRACT FROM AUTHOR]

We provide counterexamples to P. Olver’s freeness conjecture for C ∞ actions. In fact, we show that a counterexample exists for any connected real Lie group with noncompact center, as well as for the additive group of the integers.

In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples: planes, spheres, or cylinders. Generally, the characterization of such curves, both in Euclidean ( $$E^3$$ ) and in Lorentz-Minkowski ( $$E_1^3$$ ) spaces, involves an ODE relating curvature and torsion. However, by equipping a curve with a relatively parallel moving frame, Bishop was able to characterize spherical curves in $$E^3$$ through a linear equation relating the coefficients which dictate the frame motion. Here we apply these ideas to surfaces that are implicitly defined by a smooth function, $$\varSigma =F^{-1}(c)$$ , by reinterpreting the problem in the context of the metric given by the Hessian of F, which is not always positive definite. So, we are naturally led to the study of curves in $$E_1^3$$ . We develop a systematic approach to the construction of Bishop frames by exploiting the structure of the normal planes induced by the casual character of the curve, present a complete characterization of spherical curves in $$E_1^3$$ , and apply it to characterize curves that belong to a non-degenerate Euclidean quadric. We also interpret the casual character that a curve may assume when we pass from $$E^3$$ to $$E_1^3$$ and finally establish a criterion for a curve to lie on a level surface of a smooth function, which reduces to a linear equation when the Hessian is constant. [ABSTRACT FROM AUTHOR]

This paper presents path following control for fully actuated rigid body systems in 3-D space. Transverse feedback linearization and a parallel transport frame are used to design a path following controller that is suitable for paths that are parametrized as regular threefold continuously differentiable curves. The application of this frame enables the handling of paths with zero curvature and reduces the complexity of the control law. The path following controller is combined with a compliance control concept. The strategy is verified by a series of measurement results on a DELTA robot with linear drives. [ABSTRACT FROM AUTHOR]

The hyper-reduced-order model (HROM) is proposed for the thermal calculation with a constant moving thermal load. Firstly, the constant velocity transient process is simplified to a steady-state process in the moving frame. Secondly, the control volume is determined by the temperature rate, and the thermal equilibrium equation in the moving frame is derived by introducing an advective term containing the loading velocity. Thirdly, the HROM is performed on the control volume with a moving frame formulation. This HROM has been applied to the thermal loading on brick and ring disk specimens with a CPU gain of the order of 7 (107). In addition, two strategies are proposed for the HROM to improve its precision. Moreover, the high efficiency and high accuracy are kept for the parametric studies on thermal conductivity and amplitude of heat flux based on the developed HROM. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]