The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h,p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.

This paper considers dynamic behavior of non-classical thermoelastic solid continua. The mathematical model consists of the conservation and balance laws of non-classical continuum mechanics that incorporates additional physics of internal rotations arising due to deformation gradient tensor. We consider plane stress behavior with small deformation, small strain physics only. Galerkin Method with Weak Form (GM/WF) in space is considered to construct a space-time decoupled finite element formulation giving rise to ordinary differential equations (ODEs) in time containing mass matrix, stiffness matrix due to classical as well as non-classical physics and acceleration and displacement associated with nodal degrees of freedom. This formulation is utilized to: (1) study natural undamped modes of vibration ( 2) study transient dynamic response by time integrating the ODEs in time (3) study the transient dynamic response by transforming the ODEs in time to modal basis using eigenvectors of the undamped natural modes. The ODEs in modal basis are used to construct transient dynamic response by time integrating them as well as by considering their analytical solutions. The solutions of the model problem obtained using the mathematical model based on non-classical continuum mechanics with internal rotations are presented and are compared with those obtained using the mathematical model based on classical continuum mechanics to demonstrate the influence of new physics due to internal rotations on the dynamic response of solid continua. Keywords Non-classical continuum mechanics Á Lagrangian description Á Internal rotations Á Balance of moment of moments Á Natural modes of vibration Á Transient dynamic response Á Space-time coupled Á Space-time decoupled Á Normal mode synthesis In honor of Professor J. N. Reddy for his 75th Birthday.

In this paper, we present three micropolar nonclassical continuum theories (NCCT) for fluent medium in which a material point always has velocities v 𝒗 𝒗 as degrees of freedom, additionally, we consider (i) in the first NCCT, we consider classical or internal rotations rates 𝑟 𝑖 Θ 𝜣 𝜣 (known) due to skew-symmetric part of velocity gradient tensor; (ii) in the second NCCT, we consider rotation rates 𝑟 𝑖 Θ 𝜣 𝜣 and Cosserat or microrotation rates 𝑟 𝑒 Θ 𝜣 𝜣 (unknown degrees of freedom); (iii) in the third NCCT, we consider Cosserat or microrotation rates 𝑟 𝑒 Θ 𝜣 𝜣 only, hence 𝑟 𝑖 Θ 𝜣 𝜣 are neglected. Conservation and balance laws (CBL) and the constitutive theories are derived for all three NCCT. We examine consistent choice of kinematic variables, modification of the CBL of classical continuum theories (CCT) due to new physics and determine if the new physics of rotation rates requires additional balance law. All three NCCT derived here are examined for thermodynamic and/or mathematical consistency and are compared with published works. Model problem studies are also presented.

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