Hyper-dual Numbers Research Articles

IMPLEMENTATION OF THE ALGEBRA OF HYPERDUAL NUMBERS IN NEURAL NETWORKS

For the numerical solution of problems arising in various fields of mathematics and mechanics, it is often necessary to determine the values of derivatives included in the model. Currently, numerical values of derivatives can be obtained using automatic differentiation libraries in many programming languages. This paper discusses the use of the Python programming language, which is widely used in the scientific community. It should be noted that the principles of automatic differentiation are not related to numerical or symbolic differentiation methods. The work consists of three parts. The introduction reviews the historical development of the general theory of complex numbers and the use of simple complex, double and dual numbers, which are a subset of the set of general complex numbers, in various fields of mathematics. The second part is devoted to the algebra of dual and hyperdual numbers and their properties. This section presents tables of the basis element of elementary functions with dual and hyperdual arguments, based on multiplication rules. Two important formulas for finding the numerical values of a complex function's first and second derivatives by expanding functions with dual and hyperdual arguments in the Taylor series are also obtained. A simple test function was used to verify the correctness of these formulas, the results of which were checked analytically as well as through implementation in a programming language. The third part of the paper focuses on practical applications and the implementation of these methods in Python. It includes detailed examples of case studies demonstrating the effectiveness of using hyperdual numbers in automatic differentiation. The results highlight the accuracy and computational efficiency of these methods, making them valuable tools for researchers and engineers. This comprehensive approach not only validates the theoretical aspects but also showcases the practical utility of dual and hyperdual numbers in solving complex mathematical and mechanical problems.

Robust and efficient implementation of finite strain generalized continuum models for material failure: Analytical, numerical, and automatic differentiation with hyper-dual numbers

Generalized continuum models for representing nonlinear material behavior including material failure in the finite strain regime are commonly formulated based on scalar elastic and dissipation potential functions. The evolution of stresses and internal variables, i.e., the material state, is governed by partial derivatives of the potential functions with respect to deformation and stress measures. Furthermore, for application of such models in implicit nonlinear finite element analysis tangent operators, consistent with the numerical integration algorithm, are required. In this work, we study analytical, numerical and automatic differentiation schemes for the implementation of coupled, generalized continuum models considering finite inelastic deformations and fracture. This includes the application of finite difference approximations, complex-step derivative approximations and automatic differentiation based on hyper-dual numbers. For the use of automatic differentiation, a semi-analytical split approach is introduced for increasing the computational efficiency. Based on a comprehensive 2D and 3D finite element study, we demonstrate the superior properties of automatic differentiation compared to numerical differentiation methods with regards to accuracy and robustness. Furthermore, the additional computational cost for automatic differentiation using the proposed split approach becomes negligible for large problems.

A Note on Hyper-Dual Numbers with the Leonardo-Alwyn Sequence

A Note on Hyper-Dual Numbers with the Leonardo-Alwyn Sequence

Characterizations and properties of hyper-dual Moore-Penrose generalized inverse

<p>In this paper, the definition of the hyper-dual Moore-Penrose generalized inverse of a hyper-dual matrix is introduced. Characterizations for the existence of the hyper-dual Moore-Penrose generalized inverse are given, and a formula for the hyper-dual Moore-Penrose generalized inverse is presented whenever it exists. Least-squares properties of the hyper-dual Moore-Penrose generalized inverse are discussed by introducing a total order of hyper-dual numbers. We also introduce the definition of a dual matrix of order $ n $. A necessary and sufficient condition for the existence of the Moore-Penrose generalized inverse of a dual matrix of order $ n $ is given.</p>

On hyper-dual vectors and angles with Pell, Pell-Lucas numbers

<p>In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.</p>

A versatile implicit computational framework for continuum-kinematics-inspired peridynamics

Continuum-kinematics-inspired peridynamics (CPD) has been recently proposed as a novel reformulation of peridynamics that is characterized by one-, two- and three-neighbor interactions. CPD is geometrically exact and thermodynamically consistent and does not suffer from zero-energy modes, displacement oscillations or material interpenetration. In this manuscript, for the first time, we develop a computational framework furnished with automatic differentiation for the implementation of CPD. Thereby, otherwise tedious analytical differentiation is automatized by employing hyper-dual numbers (HDN). This differentiation method does not suffer from round-off errors, subtractive cancellation errors or truncation errors and is thereby highly stable with superb accuracy being insensitive to perturbation values. The computational framework provided here is compact and model-independent, thus once the framework is implemented, any other material model can be incorporated via modifying the potential energy solely. Finally, to illustrate the versatility of our proposed framework, various potential energies are considered and the corresponding material response is examined for different scenarios.

Min-Max Optimal Control of Robot Manipulators Affected by Sensor Faults.

This paper is concerned with the control law synthesis for robot manipulators, which guarantees that the effect of the sensor faults is kept under a permissible level, and ensures the stability of the closed-loop system. Based on Lyapunov's stability analysis, the conditions that enable the application of the simple bisection method in the optimization procedure were derived. The control law, with certain properties that make the construction of the Lyapunov function much easier-and, thus, the determination of stability conditions-was considered. Furthermore, the optimization problem was formulated as a class of problem in which minimization and maximization of the same performance criterion were simultaneously carried out. The algorithm proposed to solve the related zero-sum differential game was based on Newton's method with recursive matrix relations, in which the first- and second-order derivatives of the objective function are calculated using hyper-dual numbers. The results of this paper were evaluated in simulation on a robot manipulator with three degrees of freedom.

The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices

The Lie group over the hyper-dual matrices and its corresponding Lie algebra are first introduced in this study. One of Euler's strategies called the Euler-Rodrigues formula is applied to the matrices of hyper-dual rotations. The fundamental relationship between the hyper-dual numbers and the dual numbers allows us to apply the formula on dual lines and two intersecting real lines in the three-dimensional dual and Euclidean spaces, respectively.

The method for solving topology optimization problems using hyper-dual numbers

The method for solving topology optimization problems using hyper-dual numbers

Hyper-dual number-based numerical differentiation of eigensystems

Considering application to nonlinear material models, this study proposes a novel numerical method for computing arbitrary-order derivatives of eigenvalues and eigenvectors (eigensystems) for second-order tensors (matrices) based on hyper-dual numbers (HDNs). This study introduces two numerical differentiation approaches for eigensystems: one obtains derivatives of both eigenvalues and eigenvectors by inductively solving linear equations derived from an HDN formulation of eigensystem, and the other specializes in solving derivatives of a function represented by eigenvalues, such as the Helmholtz free energy function in the Ogden model, a representative example of the eigenvalue-based hyperelastic material models. Because these approaches differ considerably in applicability and computational cost, this study proposes a numerical differentiation method that uses both complementarily. The proposed method can accurately compute higher-order derivatives even if the second-order tensors have multiple solutions for eigenvalues. Moreover, applying the proposed method to an incremental variational formulation (IVF) leads to robust and accurate computations for the Ogden material model. These results imply that IVF facilitates the implementation of several constitutive models in a unified manner, and the proposed method broadens the scope of these models, including eigensystem expressions.

Kinematic Applications of Hyper-Dual Numbers

Hyper-dual numbers are a new number system that is an extension of dual numbers. A hyper-dual number can be written uniquely as an ordered pair of dual numbers. In this paper, some basic algebraic properties of hyper-dual numbers are given using their ordered pair representaions of dual numbers. Moreover, the geometric interpretation of a unit hyper-dual vector is given in module as a dual line. And a geometric interpretation of a subset of unit hyper-dual sphere (the set of all unit hyper-dual vectors) is given as two intersecting perpendicular lines in 3-dimensional real vector space.

Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling

The calculation of derivatives is ubiquitous in science and engineering. In thermodynamics, in particular, state properties can be expressed as derivatives of thermodynamic potentials. The manual differentiation of complex models can be tedious and error-prone. In this work, we revisit dual and hyper-dual numbers for the calculation of exact derivatives and show generalizations to higher order derivatives and derivatives with respect to vector quantities. The methods described in this paper are accompanied by an open source Rust implementation with Python bindings. Applications of the generalized (hyper-) dual numbers are given in the context of equation of state modeling and the calculation of critical points.

Multiple bifurcation paths visualized by a computational asymptotic stability theory

Multiple bifurcation (MB) is a compound stability problem of nonlinear structures, in which the singular system stiffness matrix at the stability point coincidentally undergoes two or more zero eigenvalues. The corresponding critical eigenvectors are generally coupled in the actual buckling modes, as frequently observed in symmetric structures.This paper presents a practical diagnosis to visualize all secondary paths branching from the compound stability point when the stiffness matrix is deficient in rank by two. To find post-critical equilibria around an MB point (MBP), asymptotic expansions are assumed to consist of two homogeneous and one particular solution of the singular stiffness equations. Furthermore, hyper-dual numbers are introduced to numerically evaluate the derivatives of the system stiffness with respect to the nodal degrees-of-freedom. The resulting bifurcation equations are a set of three simultaneous cubic polynomial equations with unknown perturbation parameters that can be solved by a popular graphical software. The number and location of existing equilibria above and beneath the MBP at the load level can exactly indicate each type of branching path, such as asymmetric, unstable, or stable symmetric bifurcation paths.Two numerical examples demonstrate that the proposed asymptotic theory can reliably diagnose MB. The first one is the Augusti model, i.e., a simple rigid column supported by elastic springs that shows that the numerical prediction by the proposed method is consistent with Augusti’s analytical results. The second example was computed using the plate and shell finite element (FE) program to verify that the asymptotically expanded and visually solved bifurcation equations work well and can be implemented in existing FE codes for stability analysis, including imperfection-sensitivity and optimization.

A study on the computational effort of hyper-dual numbers to evaluate derivatives in geometrically nonlinear hyperelastic trusses

We report the use of hyper-dual numbers as a derivative tool for stress and tangent modulus calculation in hyperelastic models. By using the analytical expressions of Ogden’s model as a reference, we confirmed the accuracy of this tool for both first- and second-order derivatives. Moreover, the hyper-dual implementation considering the operator overloading technique corroborated the interesting generality properties of this method; by using virtually the same code syntax, the material behavior was entirely described by the hyper-dual representation of the strain energy function. Therefore, the use of the hyper-dual procedure brings an important advantage in this field since the development of the expressions concerning stress and tangent modulus becomes unnecessary; only the potential function of a given model is implemented. In this context, due to its specific set of arithmetic operations, a detailed study on the efficiency of the hyper-dual scheme in a finite element analysis is demanded. In this research, we proposed a comparative study regarding the effects of the hyper-dual procedure on the analysis processing time. Using a specific mesh generator, we evaluated a wide range of model sizes for a beam structure made of hyperelastic trusses. As a result, when compared to an ordinary finite element execution using analytical expressions for the constitutive model, we found corresponding hyper-dual performance in the larger models. Particularly, we identified the model size from which the hyper-dual approach becomes competitive; considering this element type, this model contains approximately 30,000 elements and 25,000 degrees of freedom. Furthermore, as for the computational time related to the material subroutine and the stiffness matrix, we found that the hyper-dual scheme increased the computational time by a factor of 4 and 2, respectively. We demonstrated that the hyper-dual procedure combines interesting characteristics in terms of accuracy, generality and computational costs. Hence, this numerical strategy for derivative calculation potentially represents an important tool for the development and application of new constitutive models in structural mechanics.

Hyper-dual split quaternions and rigid body motion

In this paper, we introduce and define hyper-dual split vectors by using hyper-dual numbers. We define three subsets; two of them are subsets of Lorentzian unit hyper-dual sphere and the other is a subset of hyperbolic unit hyper-dual sphere. We show that to each element of these subsets corresponds two intersecting perpendicular lines in Minkowski space. We also introduce and define hyper-dual split quaternions with their basic algebraic properties. We give the geometric interpretation of hyper-dual split quaternions, and we define three new operators in Minkowski space. We show that these operators turn each two intersecting perpendicular lines again to two intersecting perpendicular lines in Minkowski space. Examples are also provided to illustrate our theorems and results.

Hyper Dual Quaternions representation of rigid bodies kinematics

Hyper-Dual Quaternions, HDQ, are first introduced in this paper. Utilizing the Hyper-Dual Numbers, HDN, Dual Quaternions, DQ, and hyper-dual angle, the general expression of HDQ is obtained. Rigid single- and multi-body kinematics such as in a serial robot kinematics, are then expressed in HDQ form. Taking advantage of the dual numbers’ “automatic differentiation” feature, HDQ encompasses both body pose (translational and rotational) and body velocities in a compact form with no need for further pose differentiation.

Hill-top branching: Its asymptotically expanded and visually solved bifurcation equations

Hill-top branching (HTB) is an exceptional stability problem in which the bifurcation point (BP) coincides exactly with a limit point (LP). In such cases, snap-through and branching instability occur simultaneously. Then, HTB is often obscured in snap-through behavior and is often overlooked in stability design. The compound instability is in fact not well realized in the community of computational mechanics, when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined. No report of the relevant literature has established diagnosis of path branching or proposed a branch-switching procedure. Furthermore, few HTB models are available in the literature.This report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue. There might be a single or two critical eigenvectors of the singular stiffness matrix. Higher-derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper-dual numbers (HDNs). The resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as MATLAB®. From the locations of existing real roots displayed on a graphic monitor, the crossover situation of path branching is visually clear above and below the hill-top. For the present study, three HTB examples are computed to verify the proposed algorithm strategy using HDNs and graphical tools. The obtained numerical results show excellent agreement with analytical and finite element predictions. The proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric HTB in structural stability problems and function well in computational engineering problems.

Formulation for an Ogden-type hyperelastic analysis with hyper dual numbers and its performance evaluation

Formulation for an Ogden-type hyperelastic analysis with hyper dual numbers and its performance evaluation

Fundamental theorem of matrix representations of hyper-dual numbers for computing higher-order derivatives

Fundamental theorem of matrix representations of hyper-dual numbers for computing higher-order derivatives

Three-dimensional phase field modeling of inhomogeneous gas-liquid systems using the PeTS equation of state.

Recently, an equation of state (EoS) for the Lennard-Jones truncated and shifted (LJTS) fluid has become available. As it describes metastable and unstable states well, it is suited for predicting density profiles in vapor-liquid interfaces in combination with density gradient theory (DGT). DGT is usually applied to describe interfaces in Cartesian one-dimensional scenarios. In the present work, the perturbed LJ truncated and shifted (PeTS) EoS is implemented into a three-dimensional phase field (PF) model which can be used for studying inhomogeneous gas-liquid systems in a more general way. The results are compared with the results from molecular dynamics simulations for the LJTS fluid that are carried out in the present work and good agreement is observed. The PF model can therefore be used to overcome the scale limit of molecular simulations. A finite element approach is applied for the implementation of the PF model. This requires the first and second derivatives of the PeTS EoS which are calculated using hyper-dual numbers. Several tests and examples of applications of the new PeTS PF model are discussed.