Scaled boundary isogeometric analysis for 2D elastostatics

A new numerical method,scaled boundary isogeometric analysis(SBIGA)combining the concept of the scaled boundary finite element method(SBFEM)and the isogeometric analysis(IGA),is proposed in this study for 2D elastostatic problems with both homogenous and inhomogeneous essential boundary conditions.Scaled boundary isogeometric transformation is established at a specified scaling center with boundary isogeometric representation identical to the design model imported from CAD system,which can be automatically refined without communication with the original system and keeping geometry invariability.The field variable,that is,displacement,is constructed by the same basis as boundary isogeometric description keeping analytical features in radial direction.A Lagrange multiplier scheme is suggested to impose the inhomogeneous essential boundary conditions.The new proposed method holds the semi-analytical feature inherited from SBFEM,that is,discretization only on boundaries rather than the entire domain,and isogeometric boundary geometry from IGA,which further increases the accuracy of the solution.Numerical examples,including circular cavity in full plane,Timoshenko beam with inhomogenous boundary conditions and infinite plate with circular hole subjected to remotely tension,demonstrate that SBIGA can be applied efficiently to elastostatic problems with various boundary conditions,and powerful in accuracy of solution and less degrees of freedom(DOF)can be achieved in SBIGA than other methods.

A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

The application of the singular boundary method(SBM),a relatively new meshless boundary collocation method,to the inverse Cauchy problem in threedimensional(3D)linear elasticity is investigated.The SBM involves a coupling between the non-singular boundary element method(BEM)and the method of fundamental solutions(MFS).The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS.Due to the boundary-only discretizations and its semi-analytical nature,the method can be viewed as an ideal candidate for the solution of inverse problems.The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique,while the optimal regularization parameter is determined by the L-curve criterion.Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.

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.

Singular integrals in axisymmetric problems of elastostatics

Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper.These equations are received after applying the integral transformation and Gauss-Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium.Initially,three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate.So,the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only.The thorough analysis of both displacement and traction kernels is accomplished,and similarity in behavior of both kernels is established.The kernels are expressed in terms of complete elliptic integrals of first and second kinds.The second kind elliptic integrals are nonsingular,and standard Gaussian quadratures are applied for their numerical evaluation.Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities.The numerical treatment of these integrals takes into account the presence of this integrable singularity.The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity,Catalan’s constant,the Gaussian surface integral.The comparison between analytical and numerical data has proved high precision and availability of the proposed method.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ)—NOETHER'S THEOREM

The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether's theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.

Some Remarks on the Method of Fundamental Solutions for Two- Dimensional Elasticity

In this paper,some remarks for more efficient analysis of two-dimensional elastostatic problems using the method of fundamental solutions are made.First,the effects of the distance between pseudo and main boundaries on the solution are investigated and by a numerical study a lower bound for the distance of each source point to the main boundary is suggested.In some cases,the resulting system of equations becomes ill-conditioned for which,the truncated singular value decomposition with a criterion based on the accuracy of the imposition of boundary conditions is used.Moreover,a procedure for normalizing the shear modulus is presented that significantly reduces the condition number of the system of equations.By solving two example problems with stress concentration,the effectiveness of the proposed methods is demonstrated.

Axisymmetric smooth contact for an elastic isotropic infinite hollow cylinder compressed by an outer rigid ring with circular profile

A contact problem for an infinitely long hollow cylinder is considered. The cylinder is compressed by an outer rigid ring with a circular profile. The material of the cylinder is linearly elastic and isotropic. The extent of the contact region and the pressure distribution are sought. Governing equations of the elasticity theory for the axisymmetric problem in cylindrical coordinates are solved by Fourier transforms and general expressions for the displacements are obtained. Using the boundary conditions, the formulation is reduced to a singular integral equation. This equation is solved by using the Gaussian quadrature. Then the pressure distribution on the contact region is determined. Numerical results for the contact pressure and the distance characterizing the contact area are given in graphical form.

Design of Calibration Experiments for Identification of Manipulator Elastostatic Parameters

The paper is devoted to the elastostatic calibration of industrial robots, which is used for precise machining of large-dimensional parts made of composite materials. In this technological process, the interaction between the robot and the workpiece causes essential elastic deflections of the manipulator components that should be compensated by the robot controller using relevant elastostatic model of this mechanism. To estimate parameters of this model, an advanced calibration technique is applied that is based on the non-linear experiment design theory, which is adopted for this particular application. In contrast to previous works, it is proposed a concept of the user-defined test-pose, which is used to evaluate the calibration experiments quality. In the frame of this concept, the related optimization problem is defined and numerical routines are developed, which allow generating optimal set of manipulator configurations and corresponding forces/torques for a given number of the calibration experiments. Some specific kinematic constraints are also taken into account, which insure feasibility of calibration experiments for the obtained configurations and allow avoiding collision between the robotic manipulator and the measurement equipment. The efficiency of the developed technique is illustrated by an application example that deals with elastostatic calibration of the serial manipulator used for robot-based machining.

Three-Dimensional Meshfree Analysis of Interlocking Concrete Blocks for Step Seawall Structure

This study adapts the flexible characteristic of meshfree method in analyzing three-dimensional(3D)complex geometry structures,which are the interlocking concrete blocks of step seawall.The elastostatic behavior of the block is analysed by solving the Galerkin weak form formulation over local support domain.The 3D moving least square(MLS)approximation is applied to build the interpolation functions of unknowns.The pre-defined number of nodes in an integration domain ranging from 10 to 60 nodes is also investigated for their effect on the studied results.The accuracy and efficiency of the studied method on 3D elastostatic responses are validated through the comparison with the solutions of standard finite element method(FEM)using linear shape functions on tetrahedral elements and the well-known commercial software,ANSYS.The results show that elastostatic responses of studied concrete block obtained by meshfree method converge faster and are more accurate than those of standard FEM.The studied meshfree method is effective in the analysis of static responses of complex geometry structures.The amount of discretised nodes within the integration domain used in building MLS shape functions should be in the range from 30 to 60 nodes and should not be less than 20 nodes.