Reduced Integration Rules Research Articles

A robust penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches in large deformations

Isogeometric Kirchhoff–Love elements have been receiving increasing attention in geometrically nonlinear analysis of thin shells because they make it possible to meet the C1 requirement in the interior of surface patches and to avoid the use of finite rotations. However, engineering structures of appreciable complexity are typically modeled using multiple patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. Simple penalty approaches for coupling adjacent patches, applicable to either smooth or non-smooth interfaces and either matching or non-matching discretizations, have been proposed. Although the problem dependence of the penalty coefficient can be reduced by scaling factors which take into account geometrical and material parameters, only high values of the penalty coefficient can guarantee a negligible coupling error in all possible cases. However, this can lead to an ill conditioned problem and to an increasing iterative effort for solving the nonlinear discrete equations. In this work, we show how to avoid this drawback by rewriting the penalty terms in an Hellinger–Reissner form, introducing independent fields work-conjugated to the coupling equations. This technique avoids convergence problems, making the analysis robust also for very high values of the penalty coefficient, which can be then employed to avert coupling errors. Moreover, a proper choice of the basis functions for the new fields provides an accurate coupling also for general non-matching cases, preventing overconstrained solutions. The additional variables are condensed out and then not involved in the global system of equations to be solved. A highly efficient approach based on a mixed integration point strategy and an interface-wise reduced integration rule makes the condensation inexpensive preserving the sparsity of the condensed stiffness matrix and the coupling accuracy.

Impact analysis of PEEK/ceramic/gelatin composite for finding behind the armor trauma

A ceramic-based armor system must arrest bullet penetration and dissipate a large amount of impact energy. Taking the total linear momentum transferred to the wearer of a ceramic armor as a measure of the behind-armor ballistic/blunt trauma (BABT), we study the effect on the BABT of adding a thin polyether-ether-ketone (PEEK) layer on the face of a Silicon Carbide (SiC) ceramic target impacted at normal incidence by a full metal jacket bullet traveling at a low velocity. A human torso is simulated by a block of ballistic gelatin. Three-dimensional non-linear large deformations of the system are numerically analyzed using the commercial finite element software ABAQUS/Explicit with the full integration rule to evaluate elemental matrices. Sensitivity tests are conducted with random samples generated by the Latin Hypercube Sampling technique to delineate the influence of the static yield stress, strain hardening, Young’s modulus and the mass density of the PEEK layer as well as of the shear modulus, the mass density and the maximum hydrostatic tensile stress of the ceramic on the BABT. It is found that reducing the acoustic mismatch between the PEEK and the ceramic increased the plastic energy dissipated in the ceramic. For normal impact at 180 m/s of a bullet having kinetic energy of 130 J, the 2 mm thick surface layer decreased the total peak normal force, the normal impulse transmitted, and the maximum pressure transferred from the SiC target to the gelatin layer by 19%, 18% and 32%, respectively. Simulations using a reduced integration rule significantly under-predict the final kinetic energy and the back face displacement of the bullet as compared to those using the full integration rule. For the 2 mm thick PEEK layer located on the top, the middle and the bottom of the SiC plate, the back face displacement is the least for the layer located in the middle but the BABT increased.

Reduced integration at superconvergent points in isogeometric analysis

We propose a new reduced integration rule for isogeometric analysis (IGA) based on the concept of variational collocation. It has been recently shown that, if a discrete space is constructed by smooth and pointwise non-negative basis functions, there exists a set of points – named Cauchy–Galerkin (CG) points – such that collocation performed at these points can reproduce the Galerkin solution of various boundary value problems exactly. Since CG points are not known a-priori, estimates are necessary in practice and can be found based on superconvergence theory. In this contribution, we explore the use of estimated CG points (i.e. superconvergent points) as numerical quadrature points to obtain an efficient and stable reduced quadrature rule in IGA. We use the weighted residual formulation as basis for our new quadrature rule, so that the proposed approach can be considered intermediate between the standard (accurately integrated) Galerkin variational formulation and the direct evaluation of the strong form in collocation approaches. The performance of the method is demonstrated by several examples. For odd degrees of discretization, we obtain spatial convergence rates and accuracy very close to those of accurately integrated standard Galerkin with a quadrature rule of two points per parametric direction independently of the degree.

New linear and quadratic prismatic piezoelectric solid–shell finite elements

In this work, we propose two prismatic piezoelectric solid–shell elements based on fully three-dimensional kinematics. For this purpose, we perform electromechanical coupling, which consists in adding an electrical degree of freedom to each node of the purely mechanics-based versions of these elements. To increase efficiency, these geometrically three-dimensional elements are provided with some desirable shell features, such as a special direction, designated as the thickness, along which the integration points are located, while adopting a reduced integration rule in the other directions. To assess the performance of the proposed piezoelectric solid–shell elements, a variety of benchmark tests, both in static and vibration analysis, have been performed on multilayer structures ranging from simple beams to more complex structures involving geometric nonlinearities. Compared to conventional finite elements with the same kinematics, the evaluation results allow highlighting the higher performance of the newly developed solid–shell technology.

Selective and reduced numerical integrations for NURBS-based isogeometric analysis

We propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on NURBS elements. The notion of an approximation space that approximates the target space is introduced. We explore the use of various approximation spaces associated with optimal patch-wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points. Patch rules exploit the higher continuity of spline basis functions. The tendency of smooth spline functions to exhibit numerical locking in nearly-incompressible problems when using a full Gauss–Legendre quadrature is alleviated with selective or reduced integration. Stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem. We propose a local algorithm, which is robust and computationally efficient, to compute element-by-element the quadrature points and weights in patch rules. The performance of the methods is assessed on several numerical examples in two-dimensional elasticity and Reissner–Mindlin shell structures.

UNCONDITIONALLY STABLE COLLOCATION ALGORITHMS FOR SECOND ORDER INITIAL VALUE PROBLEMS

In this paper, unconditionally stable higher order accurate time step integration algorithms suitable for second order initial value problems in collocation form are presented. The second order equations are manipulated directly. If the approximate solution is expressed as a polynomial of degree n+1, there are n unknowns to be determined after taking into account the two given initial conditions. It is well known that by suppressing the residuals of the governing equations at n distinct collocation points only, the resultant algorithms are only conditionally stable. In this paper, linear combinations of the residuals at n+1 distinct collocation points are used to solve for the n unknowns. The collocation points and the relative weights between the residuals are derived from the weighted residual method. The weighting functions are arbitrary polynomials of degree not exceeding n−1. To control the accuracy and stability properties of the resultant algorithms, the reduced integration technique is used to evaluate the integrals in the formulation. Once the reduced integration rules are decided, the equivalent collocation form can be derived. It is found that the resultant algorithms cast in the collocation form are easy to implement and can be used to tackle non-linear problems directly. Numerical examples are given to illustrate the validity of the present formulation.

An accurate and efficient shell element with improved reduced integration rules

An accurate and efficient shell element is presented. The stiffness of the shell element is decomposed into two parts with one part corresponding to stretching and bending deformation and the other part corresponding to shear deformation of the shell. Both parts of the stiffness are calculated with reduced integration rules, thereby improving computational efficiency. Shear strains are averaged on the reference surface such that neither locking phenomena nor any zero energy mode can occur. The satisfactory behaviour of the element is demonstrated in several numerical examples.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory

A least-squares-based pressure projection method is proposed for the nonlinear analysis of nearly incompressible hyperelastic materials. The strain energy density function is separated into distortional and dilatational parts by the use of Penn’s invariants such that the hydrostatic pressure is solely determined from the dilatational strain energy density. The hydrostatic pressure and hydrostatic pressure increment calculated from displacements are projected onto appropriate pressure fields through the least-squares method. The method is applicable to lower and higher order elements and the projection procedures can be implemented into the displacement based nonlinear finite element program. By the use of certain pressure interpolation functions and reduced integration rules in the pressure projection equations, this method can be degenerated to a nonlinear version of the selective reduced integration method.

Stiffness matrix integration rules for contact-friction finite elements

Penalty contact finite elements are effective for modeling discontinuities in finite element analyses. A key ingredient in the evaluation of contact element matrices is numerical integration. Computational experience has shown that use of full integration sometimes gives rise to spurious, annoying oscillations in contact stresses, while some reduced integration rules do not. In this paper we thoroughly investigate the effects of numerical integration on element behavior by comparison to an equivalent mixed method. Conclusions are drawn regarding desirable characteristics that an integration rule should have and an effective local stress averaging scheme for use at contact interelement boundaries is developed.

Reduced numerical integration in thermal transient finite element analysis

The stability and accuracy of the so-called “ga-family”, two level time integration algorithms for thermal transient problems are considered. In particular, the influence on the solution characteristics of the order of numerical integration rule employed in the evaluation of the element matrices is examined. Consideration is restricted to the four node bilinear and eight node biquadratic isoparametric elements. The time integration schemes considered in detail are the Crank Nicolson, Galerkin, Modified Galerkin and Euler Backward algorithms. The stability of each process is related to both the eigenproperties of the conductivity and heat capacity matrices and to the eigenvalue spectrum of the thermal equation system. It is shown that inaccuracies and instabilities can arise in solution which are associated with the poor prediction of the element properties by reduced numerical integration rules. For eight node biquadratic elements, the source of error is the inadequate modelling of the eigenvalue spectrum of the heat capacity matrix by reduced quadrature; with both full and reduced integration rules providing very similar values for the conductivity (or stiffness matrix). However, for four node bilinear elements, reduced quadrature can result in an inadequate representation of both the conductivity and heat capacity matrices. It is shown that these inaccuracies are confined to the higher thermal frequencies which can be dominant in the initial stages of a transient response and cause deterioration of the numerical solution. Therefore an “adaptive” scheme is proposed in which a full integration order is used for the first few timesteps, after which a reduced integration rule is employed. Recommendations are made for the selection of time integration algorithms, quadrature rules and element dimensioning which are substantiated by the solution of several numerical examples.

Finite element methods for constrained problems in elasticity

AbstractSeveral different variational formulations of boundary‐value problems with constraints are discussed, with particular reference to constrained problems in elasticity. Special attention is given to exterior penalty methods. A discussion of the conditions necessary for penalty methods to provide a basis for stable and convergent finite element methods is given. In particular, the use of reduced integration is discussed and criteria on the order of reduced integration rules sufficient to produce stable and convergent schemes are described. Applications of reduced integration‐penalty methods to incompressible elasticity problems and contact problems are described.