A family of C1C^1 quadrilateral finite elements

We present a novel family of $C^1$ quadrilateral finite elements, which define global $C^1$ spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree $p\geq 6$, to all degrees $p \geq 3$. Thus, we call the family of $C^1$ finite elements Brenner-Sung quadrilaterals. The proposed $C^1$ quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose $C^2$ continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~$5$, and the polynomial degrees $p=3$ and $p=4$ by employing a splitting into $3\times3$ or $2\times2$ polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree $p$. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for $p\in\{3,4,5\}$ explicit formulas for the B\'{e}zier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed $C^1$ quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for $p=5$) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom

A Comparison of Morphemic Analysis and Whole Word Meaning Instruction on Sixth-Grade Students' Knowledge of Prefixes, Taught Words, and Transfer Words

An eight-day instructional vocabulary study was conducted to evaluate two methods of instruction for prefixed words for two methods, Morphemic Analysis and Whole Word Meaning. Seventy-five sixth-grade students from a rural middle school were part of this study. The Morphemic Analysis and Whole Word Meaning approaches were similar in a number of ways. Instruction consisted of eight lessons, six instructional lessons and two review lessons. Methods were similar in the specific prefixed words taught (24), duration (8 days/8-9 minutes per word), number of exposures (9), and inclusion of the following activities: Example and/or Non-example, Student Examples, Word/Meaning Match, and Word/Example Match. The major differences between the two methods occurred during the introduction of the prefixed words. Morphemic Analysis included a prefix component that focused on grouping prefixes by families, introducing each prefix meaning, and then analyzing the prefixed word by morphemes: root, prefix, and suffix (as needed). The meaning of the prefixed word was derived by combining the meanings of the parts: root, prefix, and suffix. Whole Word Meaning instruction focused on the prefixed word as a whole unit. Meaning for the prefixed word was developed from a Scenario and Question activity. This activity placed the lesson word into a meaningful written context, and a question followed that guided students to infer the word's meaning. Also, a Prompt activity was used to extend the word's meaning beyond the written passage.Analysis of data on the following three measures: 24 prefixes, 24 prefixed lesson words, and 24 untaught prefixed words, revealed students' performance for the two conditions, Morphemic Analysis and Whole Word Meaning. The data revealed that students made a greater gain in prefix knowledge (17%) from Morphemic Analysis instruction. This gain could be attributed to the direct instruction of prefixes, a major component of the Morphemic Analysis method. On prefixed lesson words, Morphemic Analysis and Whole Word Meaning each showed large gains; thus, they could be considered equally effective methods of vocabulary instruction. The data on untaught prefixed words indicated that the Morphemic Analysis group outperformed the Whole Word Meaning group, by an advantage of two mean points (8%). The present study points to the benefits of prefix knowledge and transfer word knowledge for the Morphemic Analysis group. The similar performance by both methods on taught prefixed words was equally interesting and warrants further investigation into the components of effective vocabulary instruction

Assessment of Preliminary Design Approaches for Metallic Stiffened Cylindrical Shell Instability Problems

A preliminary design tool for metallic stiffened fuselage cylindrical panels subjected to longitudinal compression has been developed and validated by comparison to test results. Several methodologies for stiffened panel buckling and failure predictions were examined and evaluated. An appropriate level of analysis fidelity was determined for different failure modes and design details. Results from panel tests conducted to verify analytical methods used to design the Gulfstream V aircraft were presented. The panels were representative of four general skin/stringer configurations on the aircraft. Finite Element analyses and standard analytical methods were used to predict panel failure loads. The accuracy of the finite element analysis predictions was dependent upon the level of detail included in the model. The inclusion of such details as fasteners had a significant effect on the predicted failure load. The omission of such complexities from the finite element model led to unconservative failure predictions. Standard analytical methods were found to be more efficient than finite element methods and produced conservative panel failure loads. Improvements for a preliminary design tool were identified to reduce conservatism in failure predictions and thereby reduce structural weight

Algorithmic Foundations of Heuristic Search using Higher-Order Polygon Inequalities

The shortest path problem in graphs is both a classic combinatorial optimization problem and a practical problem that admits many applications. Techniques for preprocessing a graph are useful for reducing shortest path query times. This dissertation studies the foundations of a class of algorithms that use preprocessed landmark information and the triangle inequality to guide A* search in graphs. A new heuristic is presented for solving shortest path queries that enables the use of higher order polygon inequalities. We demonstrate this capability by leveraging distance information from two landmarks when visiting a vertex as opposed to the common single landmark paradigm. The new heuristic’s novel feature is that it computes and stores a reduced amount of preprocessed information (in comparison to previous landmark-based algorithms) while enabling more informed search decisions. We demonstrate that domination of this heuristic over its predecessor depends on landmark selection and that, in general, the denser the landmark set, the better heuristic performs. Due to the reduced memory requirement, this new heuristic admits much denser landmark sets. We conduct experiments to characterize the impact that landmark configurations have on this new heuristic, demonstrating that centrality-based landmark selection has the best tradeoff between preprocessing and runtime. Using a developed graph library and static information from benchmark road map datasets, the algorithm is compared experimentally with previous landmark-based shortest path techniques in a fixed-memory environment to demonstrate a reduction in overall computational time and memory requirements. Experimental results are evaluated to detail the significance of landmark selection and density, the tradeoffs of performing preprocessing, and the practical use cases of the algorithm