Frame-invariance in finite element formulations of geometrically exact rods

This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE（3）, or I is SE（3）-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element for- mulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equiv- ariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this al- gorithm.

On the geometric phase in the spatial equilibria of nonlinear rods

Geometric phases have natural manifestations in large deformations of geometrically exact rods. The primary concerns of this article are the physical implications and observable consequences of geometric phases arising from the deformed patterns exhibited by a rod subjected to end moments. This mechanical problem is classical and has a long tradition dating back to Kirchhoff. However, the perspective from geometric phases seems to go more deeply into relations between local strain states and global geometry of shapes, and infuses genuinely new insights and better understanding, which enable one to describe this kind of deformation in a neat and elegant way. On the other hand, visual representations of these deformations provide beautiful illustrations of geometric phases and render the meaning of the abstract concept of holonomy more direct and transparent.

Using ultrasound to improve the sequential post-synthesis modification method for making mesoporous Y zeolites

Mesoporous Y zeolites were prepared by the sequential chemical dealumination(using chelating agents such as ethylenediaminetetraacetic acid,H EDTA,and citric acid aqueous solutions)and alkaline desilication(using sodium hydroxide,NaOH,aqueous solutions)treatments.Specifically,the ultrasound-assisted alkaline treatment(i.e.,ultrasonic treatment)was proposed as the alternative to conventional alkaline treatments which are performed under hydrothermal conditions.In comparison with the hydrothermal alkaline treatment,the ultrasonic treatment showed the comparatively enhanced efficiency(with the reduced treatment time,i.e.,5 min vs.30 min,all with 0.2 mol·L NaOH at 65°C)in treating the dealuminated Y zeolites for creating mesoporosity.For example,after the treatment of a dealuminated zeolite Y(using 0.1 mol·L H EDTA at 100°C for 6 h),the ultrasonic treatment produced the mesoporous zeolite Y with the specific external surface area()of 160 m·g and mesopore volume()of 0.22 cm·g,being slightly higher than that by the conventional method(i.e.,=128 m·g and=0.19 cm·g).The acidic property and catalytic activity(in catalytic cracking of-octane)of mesoporous Y zeolites obtained by the two methods were comparable.The ultrasonic desilication treatment was found to be generic,also being effective to treat the dealuminated Y zeolites by citric acid.Additionally,the first step of chemical dealumination treatment was crucial to enable the effective creation of mesopores in the parent Y zeolite(with a silicon-to-aluminium ratio,Si/Al=2.6)regardless of the subsequent alkaline desilication treatment(i.e.,ultrasonic or hydrothermal).Therefore,appropriate selection of the condition of the chemical dealumination treatment based on the property of parent zeolites,such as Si/Al ratio and crystallinity,is important for making mesoporous zeolites effectively.