Additional Constraint Equations Research Articles

Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators (PMs), particularly those with over constrained subclosed loops, poses a challenge while ensuring numerical stability. This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis (MSA) and independent displacement coordinates (IDCs) extraction techniques. To begin, the closed-loop PM is transformed into an open-loop PM by eliminating constraints. A subassembly element is then introduced, which considers the flexibility of both rods and joints. This approach helps circumvent the numerical instability typically encountered with traditional constraint equations. The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix, utilizing multipoint constraint theory and singularity analysis, all unified within a single coordinate frame. Subsequently, the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix, alongside its elastostatic model. The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix, eliminating the need for additional equations. An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method. In conclusion, the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.

Algebraic 3D Graphic Statics with Edge and Vertex Constraints: A Comprehensive Approach to Extend the Solution Space for Polyhedral Form-Finding

Built upon a previous algebraic framework for polyhedron-based 3D Graphic Statics (3DGS) that can numerically solve for a variety of dual diagrams given an input force or form diagram, this paper introduces an improved algebraic formulation that integrates edge lengths and vertex location constraints for better control over the results. Those constraints are realized by additional edge and vertex constraint equations to previously established closing equations. The entire system of equations can be solved using the Moore–Penrose inverse (MPI) method, and each solution represents a set of compatible edge lengths for the dual diagram to be constructed. The whole solution space of the equation system provides a wide range of dual diagrams, including forms with both tensile and compressive members, which can be easily explored and was not possible using iterative methods or previous algebraic formulations. This improved formulation has been computationally implemented and released as part of a plug-in software program within the environment of Rhino3D® and Grasshopper3D®, enriching the structural form-finding toolset for designers, engineers, researchers, and educators. The tool’s performance and accuracy are demonstrated through a series of comparative studies with iterative methods. Various case studies are also presented to showcase the application of this method.

Явное представление сокращенных в размерности уравнений Эйлера сжимаемой жидкости и полной системы уравнений гидродинамики в интегральной форме

Various ways of reducing the complete system of hydrodynamic equations in volume to a system of equations on the surface are of great scientific interest. In this article, the unsteady Euler equations of a compressible fluid and the complete system of hydrodynamic equations are reduced to “stationary” integral-differential equations where time derivatives are absent. Such a procedure (3D → 2D, 2D → 1D) makes it possible to reduce the dimension of the problem by one, which significantly reduces the necessary computing power when modeling them. If to set the correct problem, then one can determine the entire unsteady flow in the volume without solving the unsteady problem. It is enough to set time-varying data only on some surface of this stream, which must be determined independently. The peculiarity of this work is that all equations reduced in dimension are obtained in explicit form, unlike previous works of the authors, where up to 200–500 equations with reduced dimension were proposed, which are very difficult to study and model. In previous authors’ work, dimensionality was reduced by reducing overdetermined systems of differential equations. In this method, in the case of a successful choice of an additional constraint equation for overdetermined systems of differential equations, there is a reduction to PDE systems of dimension less than that of the original PDE systems. The method that is used to reduce the dimension of the Euler equations and the complete system of hydrodynamic equations in this paper is actually a special case of the above. The technique of dimensional reduction in overdetermined systems of PDEs is obviously generalized to the case of integral-differential equations. The article also obtained integral equations in new variables (Lagrangian and pseudo-Lagrangian), which determine the evolution of the flow. Also, a new method is proposed for overdetermination of any system of PDEs using the general integral space relation, which is consequence of the Helmholtz decomposition theorem.

On the Constrained Solution of RBF Surface Approximation

In this contribution, a scattered data approximation problem, chosen from the literature and using the Radial Basis Function (RBF) approach, is considered for the application of point cloud modelling. Three solutions are investigated for the approximation problem. First, a technique known from the literature is investigated using a linear combination of thin-plate splines and linear polynomials, with additional constraint equations. Then, using the same approximation function as before, a technique is developed for a rigorous consideration of the constraint equations. Finally, a technique is presented in which the approximation function consists only of a linear combination of thin-plate splines, without the introduction of linear polynomials and constraint equations. In addition, some interpolation problems with the RBF approach are discussed to present the differences between an interpolation with thin-plate splines only and an interpolation with thin-plate splines together with linear polynomials and constraint equations. Numerical examples are given to illustrate and discuss the solutions from the different techniques.

Are there BPS dyons in the generalized SU(2) Yang–Mills–Higgs model?

We use the well-known Bogomolny’s equations, in general coordinate system, for BPS monopoles and dyons in the SU(2) Yang–Mills–Higgs model to obtain an explicit form of BPS Lagrangian density under the BPS Lagrangian method. We then generalize this BPS Lagrangian density and use it to derive several possible generalized Bogomolny’s equations, with(out) additional constraint equations, for BPS monopoles and dyons in the generalized SU(2) Yang–Mills–Higgs model. We also compute the stress–energy–momentum tensor of the generalized model, and argue that the BPS monopole and dyon solutions are stable if all components of the stress-tensor density are zero in the BPS limit. This stability requirement implies the scalar fields-dependent couplings to be related to each other by an equation, which is different from the one obtained in Atmaja and Prasetyo (Adv High Energy Phys 2018:7376534, arXiv: 1803.06122, 2018), and then picks particular generalized Bogomolny’s equations, with no additional constraint equation, out of those possible equations. We show that the computations in [1] are actually incomplete. Under the Julia–Zee ansatz, the generalized Bogomolny’s equations imply all scalar fields-dependent couplings must be constants, whose solutions are the BPS dyons of the SU(2) Yang–Mills–Higgs model (Prasad and Sommerfield in Phys Rev Lett 35:760, 1975), or in another words there are no generalized BPS dyon solutions under the Julia–Zee ansatz. We propose two possible ways for obtaining generalized BPS dyons, where at least one of the scalar fields-dependent couplings is not constant, that are by using different ansatze, such as axially symmetric ansatz for higher topological charge dyons; and/or by considering the most general BPS Lagrangian density.

Явное представление сокращенных в размерности уравнений Эйлера и Навье – Стокса несжимаемой жидкости в интегральной форме

In this paper, we have obtained “stationary” systems of integraldifferential equations, which are consequences of the non-stationary Euler and Navier - Stokes equations of an incompressible fluid, and for which there are no time derivatives. If we set a correct problem for them, then we can determine the entire non-stationary flow in the volume without solving the nonstationary problem. It is enough to specify time-varying data only on some surface of this stream. The method of reduction of overdetermined systems of differential equations, proposed earlier by the authors, is used. In this method, in the case of a successful choice of the additional constraint equation, the overdetermined systems of differential equations are reduced to PDE systems of dimension less than that of the original PDE systems. The Euler and Navier-Stokes equations themselves act as the constraint equations, and the dimension is reduced for the integral equations, which are obtained using the Helmholtz theorem on the expansion of an arbitrary vector field into vortex and potential components. The peculiarity of this work lies in the fact that all the equations reduced in dimension are obtained in an explicit form, in contrast to the previous works of the authors, where up to 200–300 equations with reduced dimensions were proposed. In reduced systems of integral-differential equations, there is integration over space, therefore, reduction over spatial variables by given method is impossible. If this method is generalized a little, then we can obtain a reduced system of integral equations, where the unknowns have no derivatives with respect to space, but integration over space remains. Non-stationary new integral equations are also obtained, which determine the evolution of the flow. In Appendices A and B, sufficient conditions for the correctness of the reduced system of integral-differential equations are derived. Conditions are found under which the Euler equations follow directly from these reduced equations. It is shown how the time-varying data on some surface of this stream should be related. The change of variables, which is used in the reduction of the Navier - Stokes equations, is also investigated he resulting integral equations can be used to study complex vortices in the atmosphere. For example, if devices measuring data on its surface are installed on a flying object, then, knowing the vorticity profile ω0(r) at a certain moment in time, it is possible, by solving these equations, to track vortex activity at a distance from this object in real time.

К ИДЕНТИФИКАЦИИ РЕШЕНИЙ УРАВНЕНИЯ РИККАТИ И ДРУГИХ ПОЛИНОМИАЛЬНЫХ СИСТЕМ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

The authors previously proposed a general method for finding particular solutions for overdetermined PDE systems, where the number of equations is greater than the number of unknown functions. The essence of the method is to reduce the PDE to systems of PDE of a lower dimension, in particular, to ODEs by overdetermining them by additional constraint equations. Reduction of some PDE systems generates overdetermined systems of polynomial ODEs, which are studied in this paper. A method for transforming polynomial ODE systems to linear ODE systems is proposed. The result is interesting from a theoretical point of view if these systems of polynomial ODEs are with constant coefficients. The solution of such nonlinear systems using our method can be represented as a sum of a very large but finite number of oscillations. The amplitudes of these oscillations depend on the initial data nonlinearly. The Navier–Stokes equations and unified PDE systems obtained by the authors earlier can be transformed to such systems. The Riccati equation is also investigated. New special cases are indicated when it is possible to find its solution. Numerical estimates of the complexity of this method for practical implementation are presented.

An efficient 3D iterative interface-correction reinitialization for the level set method

The level set (LS) method is a popular approach to represent interfaces owing to its smooth description of interfaces, accurate interface information and the convenience of applying to 3D, but it suffers from severe mass loss/gain mainly due to the reinitialization. To address this problem, an efficient 3D iterative interface-correction reinitialization for the LS method is proposed to maintain mass conservation and capture the accurate interface by keeping zero-level-set undisturbed. The standard reinitialization is executed as an outer iteration process to reconstruct LS functions and the operation of fixing the interface location is performed as an inner iteration process. The inner iteration firstly determines displacement errors of zero-level-set interpolation points and then redistribute these errors iteratively to adjacent nodes. The disturbed interface can return to its original position when the displacement errors tend to zero. This method requires only a small number of inner iterations without additional constraint equations and complex coupling operations. The novel approach exhibits excellent properties of mass conservation and interface preservation with the low-cost computing. Several benchmark examples are carried out to illustrate the accuracy and efficiency of the new method. The simulation results indicate that the proposed scheme is found to be highly reliable and accurate.

Searching for BPS vortices with nonzero stress tensor in the generalized Born–Infeld–Higgs model

In this article, we show how the new Bogomolny’s equations for BPS vortices, with nonzero stress tensor, in the three-dimensional generalized Maxwell–Higgs model can be derived rigorously using the BPS Lagrangian method. Particularly, we add into the original BPS Lagrangian, which is a total derivative term, two additional terms that are proportional to square of the first derivative of scalar effective field and to a function that depends only on the scalar effective field. These additional terms imply additional constraint equations which are the Euler–Lagrange equations of the BPS Lagrangian. We employ this procedure to the generalized Born–Infeld–Higgs model and obtain new Bogomolny’s equations for BPS vortices. We show their static energy could be finite if the scalar potential $$0<V< 2b^2$$ and $$V\ne b^2$$ , with b being the Born–Infeld parameter. We also show their stress tensor is nonzero and parameterized by a constant $$C_0$$ . It turns out the total static energy is proportional to the topological charge N and also the constant $$C_0$$ . We do the numerical analysis and find that solutions of the scalar and gauge effective fields behave nicely near the origin, but unfortunately they diverge near the boundary at the order of $$\mathcal{O}(r^4)$$ . We suggest incorporating gravity or considering BPS vortex in higher-dimensional models might resolve this problem. Although there are no regular solutions, the BPS Lagrangian method could be used to find Bogomolny’s equations, with nonzero stress tensor, for other BPS solitons whose existence might play a role in the Kibble–Zurek mechanism.

A dual interpolation Galerkin boundary face method for potential problems

A dual interpolation Galerkin boundary face method (DiGBFM) is applied in this paper by combining the newly developed dual interpolation method with the Galerkin boundary face method (BFM). The dual interpolation method unifies the conforming and nonconforming elements in the BFM implementation. It classifies the nodes of a conventional conforming element into virtual nodes and source nodes. Potentials and fluxes are interpolated using the continuous elements in the same way as conforming BFM, while boundary integral equations (BIEs) are collocated at source nodes, in the same way as nonconforming BFM. In order to arrive at a square linear system, we provide additional constraint equations, which are established by the moving least-squares (MLS) approximation, to condense the degrees of freedom relating to virtual nodes. Compared with the traditional symmetric Galerkin boundary element method (BEM), the symmetry feature of the DiGBFM equations is obtained simply through matrix manipulations, because of the use of the symmetric BEM, and no hypersingular BIE is needed in the DiGBFM. The proposed method has been implemented successfully for solving 2-D steady-state potential problems. Several numerical examples are presented in this paper to show the convergence and accuracy of this new method.

Mixed-dimensional modeling of structures with thin coatings using rotation-free elements

This paper describes a new mixed-dimensional finite element model for structures that consist of thick substrates with thin coatings. For such structures, the difference in length scale between the substrate and the coating often makes meshing problematic, leading to severe solution inefficiencies for conventional finite element techniques. In the new approach presented here, rotation-free beam elements are used to model the thin coating, and solid elements to model the substrate. The new model employs a unique transition element to couple the rotation-free beam elements to biquadratic quadrilateral elements without using additional constraint equations. The 10-node transition element is created by modifying the shape functions of a biquadratic quadrilateral element to ensure that the displacement field on the coated surface is consistent with the displacement field of an overlaid rotation-free beam element. Convergence studies of thin-coated structures with both straight and curved geometries show excellent accuracy while using orders of magnitude fewer degrees of freedom than a conventional finite element model using only solid elements. Hence the new model enables efficient analysis of complicated coated structures.

A dual interpolation boundary face method for exterior acoustic problems based on the Burton–Miller formulation

A dual interpolation boundary face method (DiBFM) based on the Burton–Miller formulation for solving exterior acoustic problems is presented. The method is able to unify the conforming and nonconforming BFM. In DiBFM implementation, the nodes of a conforming element are divided into two categories: source nodes (inside the element) and virtual nodes (on the boundary of the element). Acoustic pressure and its derivative are approximated using conforming elements, in the same way as conforming BFM. However, the Burton–Miller integral equations are just collocated at source nodes, in the same way as nonconforming BFM. To arrive at a square linear system, additional constraint equations established by the moving least-squares approximation are provided to condense the degrees of freedom relating to virtual nodes. Several numerical examples demonstrate the accuracy and efficiency of the proposed method.

Kinematics analysis of scissor-type inspired interconnects

Scissor-type inspired structure is used to design stretchable electronics, which aims to reduce local deformation of interconnects during large stretchability of stretchable electronics for better fatigue design. An additional ultra-soft grid method is developed to compute the deformation and shape transformations of the scissor-type inspired structures. It has been proved that the additional ultra-soft grid is only used to supply additional constraint equations and will not affect the deformation and shape transformations of the structures. As case study, the method can be used to compute the deformation and shape transformations of the structures. The simulation results show that the scissor-type inspired interconnects design can be effective to reduce the local maximal strain amplitude during cyclic stretching, which can be expected to improve the fatigue life of stretchable electronics devices dramatically.

BPS Equations of Monopole and Dyon in SU(2) Yang-Mills-Higgs Model, Nakamula-Shiraishi Models, and Their Generalized Versions from the BPS Lagrangian Method

We apply the BPS Lagrangian method to derive BPS equations of monopole and dyon in the SU2 Yang-Mills-Higgs model, Nakamula-Shiraishi models, and their generalized versions. We argue that, by identifying the effective fields of scalar field, f, and of time-component gauge field, j, explicitly by j=βf with β being a real constant, the usual BPS equations for dyon can be obtained naturally. We validate this identification by showing that both Euler-Lagrange equations for f and j are identical in the BPS limit. The value of β is bounded to β&lt;1 due to reality condition on the resulting BPS equations. In the Born-Infeld type of actions, namely, Nakamula-Shiraishi models and their generalized versions, we find a new feature that, by adding infinitesimally the energy density up to a constant 4b2, with b being the Born-Infeld parameter, it might turn monopole (dyon) to antimonopole (antidyon) and vice versa. In all generalized versions there are additional constraint equations that relate the scalar-dependent couplings of scalar and of gauge kinetic terms or G and w, respectively. For monopole the constraint equation is G=w-1, while for dyon it is wG-β2w=1-β2 which further gives lower bound to G as such G≥2β1-β2. We also write down the complete square-forms of all effective Lagrangians.

A dual interpolation boundary face method for elasticity problems

A dual interpolation boundary face method (DiBFM) is proposed to unify the conforming and nonconforming elements in boundary element method (BEM) implementation. In the DiBFM, the nodes of a conventional conforming element are sorted into two groups: the nodes on the boundary (called virtual nodes) and the internal nodes (called source nodes). Without virtual nodes, the conforming element turns to be a conventional nonconforming element of a lower order. Physical variables are interpolated using the conforming elements, the same way as conforming BEM. Boundary integral equations are collocated at source nodes, the same way as nonconforming BEM. To make the final system of linear equations solvable, additional constraint equations are required to condense the degrees of freedom for all the virtual nodes. These constraints are constructed using the moving least-squares (MLS). Besides, both boundary integration and MLS are performed in the parametric spaces of curves, namely, the geometric data, such as coordinates, out normals and Jacobians, are calculated directly from curves rather than from elements. Thus, no geometric errors are introduced no matter how coarse the discretization is. The method has been implemented successfully for solving two-dimensional elasticity problems. A number of numerical examples with real engineering background have demonstrated the accuracy and efficiency of the new method.

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

A double-layer interpolation method for implementation of BEM analysis of problems in potential theory

Abstract A double-layer interpolation method (DLIM) is proposed to improve the performance of the boundary element method (BEM). In the DLIM, the nodes of an element are sorted into two groups: (i) nodes inside the element, called source nodes, and (ii) nodes on the vertices and edges of the element, called virtual nodes. With only source nodes, the element becomes a conventional discontinuous element. Taking into account both source and virtual nodes, the element becomes a standard continuous element. The physical variables are interpolated by continuous elements (first-layer interpolation), while the boundary integral equations are collocated at the source nodes only. We further established additional constraint equations between source and virtual nodes using a moving least-squares (MLS) approximation (second-layer interpolation). Using these constraints, a square coefficient matrix of the overall system of linear equations was finally achieved. The DLIM keeps the main advantages of MLS, such as significantly alleviating the meshing task, while providing much better accuracy than the traditional BEM. The method has been used successfully for solving potential problems in two dimensions. Several numerical examples in comparison with other methods have demonstrated the accuracy and efficiency of our method.

Nonholonomic jet deformations, exact solutions for modified Ricci soliton and Einstein equations

Let [Formula: see text] be a pseudo-Riemannian metric of arbitrary signature on a manifold [Formula: see text] with conventional [Formula: see text]-dimensional splitting, [Formula: see text] determined by a nonholonomic (nonintegrable) distribution [Formula: see text] defining a generalized (nonlinear) connection and associated nonholonomic frame structures. We work with an adapted linear metric compatible connection [Formula: see text] and its nonzero torsion [Formula: see text], both completely determined by [Formula: see text]. Our first goal is to prove that there are certain generalized frame and/or jet transforms and prolongations with [Formula: see text] into explicit classes of solutions of some generalized Einstein equations [Formula: see text], [Formula: see text], encoding various types of (nonholonomic) Ricci soliton configurations and/or jet variables and symmetries. The second goal is to solve additional constraint equations for zero torsion, [Formula: see text], on generalized solutions constructed in explicit forms with jet variables and extract Levi-Civita configurations. This allows us to find generic off-diagonal exact solutions depending on all space time coordinates on [Formula: see text] via generating and integration functions and various classes of constant jet parameters and associated symmetries. Our third goal is to study how such generalized metrics and connections can be related by the so-called “half-conformal” and/or jet deformations of certain subclasses of solutions with one, or two, Killing symmetries. Finally, we present some examples of exact solutions constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by nonholonomic jet structures and generalized connections.

A constrained assumed modes method for solution of a new dynamic equation for an axially moving beam

Flexible beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the beam. In the present research, a horizontal flexible link sliding through a passive prismatic joint while attached to a rigid link of a robot moving in a vertical direction is considered. A body coordinate system is used which aids in obtaining a new and rather simple form of the differential equation without the loss of generality. To model the passive prismatic joint, the motion differential equation is written in a form of virtual displacement. Next, a solution method is presented for the lateral vibrations of the beam referred to as “constrained assumed modes method”. Unlike the traditional assumed modes method, in the proposed constrained assumed modes method, the assumed mode shapes do not each satisfy the geometrical boundary conditions of the point where passive prismatic joint is located. Instead, by writing additional constraint equations the combination of the assumed modes will satisfy the geometrical boundary conditions at location of the passive prismatic joint. Two case studies for the effect of axial motion on lateral vibration of the beam are presented. Approximate analytical results are compared with FEM results.

Manufacturing constraints in parameter–free shape optimization

AbstractStructural shape optimization has become an important tool for engineers when it comes to improving components with respect to a given goal function. During this process the designer has to ensure that the optimized part stays manufacturable. Depending on the manufacturing process several requirements could be relevant such as demolding or different kinds of symmetry. This work introduces two approaches on how to handle manufacturing constraints in parameter‐free shape optimization. In the so–called explicit approach equality and inequality equations are formulated using the coordinates of the FE‐nodes. These equations can be used to extend the optimization problem. Since the number of the additional constraint equations may be very large we apply aggregation formulations, e.g. the Kreisselmeier‐Steinhauser function, if necessary. In the second approach, the so–called implicit method, the set of design nodes is split in two groups called optimization nodes and dependent nodes. The optimization nodes are now handled as design nodes but the dependent nodes are coupled to the optimization nodes in such a way that the manufacturing constraint is fulfilled. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)