A New Approach to Obtaining Closed-Form Solutions for Higher Order Tetrahedral Finite Elements Using Modern Computer Algebra Systems

2011 ◽  
Author(s):  
Sara E. McCaslin ◽  
Panos S. Shiakolas ◽  
Brian H. Dennis ◽  
Kent L. Lawrence
Keyword(s):  
Parallel Processing ◽  
Closed Form ◽  
Computer Algebra ◽  
High Order ◽  
Computer Algebra Systems ◽  
Element Analysis ◽  
New Approach ◽  
Closed Form Solutions ◽  
Modern Computer ◽  
Stiffness Matrices

Closed-form solutions for straight-sided tetrahedral element stiffness matrices used in finite element analysis have been proven more efficient than numerically integrated solutions. These closed-form solutions are symbolically integrated using computer algebra systems such as Mathematica or Maple. However, even with memory and processing speed available on desktop computers today, major hindrances exist when attempting to symbolically evaluate the stiffness matrices for high order elements. This research proposes a new approach to obtaining closed-form solutions. Results are presented that demonstrate the feasibility of obtaining the stiffness matrices for high order tetrahedral elements through p-level 9 by use of parallel processing tools in Mathematica 7. Comparisons are made between serial and parallel approaches based on memory required to generate a solution. The serial approach requires more memory and can only generate closed-form solutions up to 7th order. The parallel processing approach presented requires less memory and can generate solutions up to 9th order.

Comparison of Computer Algebra Systems for Closed Form Stiffness Matrix Generation

2012 ◽  
Author(s):  
Panos S. Shiakolas ◽  
David C. Wilhite ◽  
Sara E. McCaslin
Keyword(s):  
Closed Form ◽  
Computer Algebra ◽  
Stiffness Matrix ◽  
Source Code ◽  
Computer Algebra Systems ◽  
Symbolic Form ◽  
Source Codes ◽  
Stiffness Matrices ◽  
Time Savings ◽  
Common Subexpressions

Computer algebra systems (CAS) have been advantageously employed to generate closed form expressions for finite elements. The advantages relate to the time improvements or savings realized by employing closed form generated expressions as compared to numerical integration. However as the element order increases, the size of the closed form generated expressions become unmanageable causing the source code files to possibly become unusable due to their size. One approach to reducing the size of the source files is to take advantage of the utilities found in CAS to identify common expressions or sub-expressions. In this manuscript we present on-going research by comparing two widely used CAS, Mathematica and Maple, as they relate to identifying common expressions in low order tetrahedral finite element stiffness matrices generated in symbolic form, associated time savings and possible issues. The results indicate that the use of CAS could be advantageously employed to identify common subexpressions through pattern matching to further reduce the size of the generated source files and realize time improvements during execution of the source codes. In addition, the developed procedures could be easily applied to higher order elements with much larger number of entries of closed form expressions where even more savings could be realized.

Validation of Nitinol SMA Characteristics Using Finite Element Analysis and Closed Form Solutions

2013 ◽  
Vol 856 ◽  
pp. 147-152
Author(s):  
S.H. Adarsh ◽  
U.S. Mallikarjun
Keyword(s):  
Experimental Data ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fe Analysis ◽  
Element Analysis ◽  
Complex Behaviour ◽  
Closed Form Solutions

Shape Memory Alloys (SMA) are promising materials for actuation in space applications, because of the relatively large deformations and forces that they offer. However, their complex behaviour and interaction of several physical domains (electrical, thermal and mechanical), the study of SMA behaviour is a challenging field. Present work aims at correlating the Finite Element (FE) analysis of SMA with closed form solutions and experimental data. Though sufficient literature is available on closed form solution of SMA, not much detail is available on the Finite element Analysis. In the present work an attempt is made for characterization of SMA through solving the governing equations by established closed form solution, and finally correlating FE results with these data. Extensive experiments were conducted on 0.3mm diameter NiTinol SMA wire at various temperatures and stress conditions and these results were compared with FE analysis conducted using MSC.Marc. A comparison of results from finite element analysis with the experimental data exhibits fairly good agreement.

Symmetry transformation matrices for structures

2007 ◽  
Vol 463 (2082) ◽  
pp. 1563-1583 ◽  
Author(s):  
Markus Pagitz ◽  
Jonathan James
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Symmetry Transformation ◽  
Symmetry Groups ◽  
Vector Spherical Harmonics ◽  
Alternative Derivation ◽  
Closed Form Solutions ◽  
Transformation Matrices ◽  
Displacement Vectors ◽  
Stiffness Matrices

Many structures in nature and engineering are symmetric. Depending on the degree of symmetry, it is possible to simplify the computations considerably by block diagonalizing the stiffness matrices. Closed-form solutions of transformation matrices for such block diagonalizations can be derived using group theory for arbitrary symmetry groups. This paper presents closed-form solutions of transformation matrices based on an alternative derivation. It is shown that transformation matrices for C nv and D nh groups can be obtained from a finite Fourier series decomposition of load and displacement vectors. Furthermore, it is shown that structures with tetrahedral, octahedral and icosahedral symmetries can be block diagonalized in an elegant way using vector spherical harmonics.

An Algorithm for Reducing the Size of Finite Element Closed-Form Source Code Files

2009 ◽  
Author(s):  
Sara McCaslin ◽  
Kent Lawrence
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Closed Form ◽  
Source Code ◽  
Simple Algorithm ◽  
Higher Order ◽  
Error Estimator ◽  
Element Analysis ◽  
Closed Form Solutions ◽  
Large Source

Closed-form solutions, as opposed to numerically integrated solutions, can now be obtained for many problems in engineering. In the area of finite element analysis, researchers have been able to demonstrate the efficiency of closed-form solutions when compared to numerical integration for elements such as straight-sided triangular [1] and tetrahedral elements [2, 3]. With higher order elements, however, the length of the resulting expressions is excessive. When these expressions are to be implemented in finite element applications as source code files, large source code files can be generated, resulting in line length/ line continuation limit issues with the compiler. This paper discusses a simple algorithm for the reduction of large source code files in which duplicate terms are replaced through the use of an adaptive dictionary. The importance of this algorithm lies in its ability to produce manageable source code files that can be used to improve efficiency in the element generation step of higher order finite element analysis. The algorithm is applied to Fortran files developed for the implementation of closed-form element stiffness and error estimator expressions for straight-sided tetrahedral finite elements through the fourth order. Reductions in individual source code file size by as much as 83% are demonstrated.

scholarly journals Applying Computer Algebra Systems in Approximating the Trigonometric Functions

2018 ◽  
Vol 23 (3) ◽  
pp. 37 ◽  
Author(s):  
Le Quan ◽  
Thái Nhan
Keyword(s):  
Error Analysis ◽  
Computer Algebra ◽  
Numerical Algorithms ◽  
High Accuracy ◽  
Computer Algebra Systems ◽  
Trigonometric Functions ◽  
Modern Computer ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Very High

We propose numerical algorithms which can be integrated with modern computer algebra systems in a way that is easily implemented to approximate the sine and cosine functions with an arbitrary accuracy. Our approach is based on Taylor’s expansion about a point having a form of kp, k∈Z and p=π/2, and being chosen such that it is closest to the argument. A full error analysis, which takes advantage of current computer algebra systems in approximating π with a very high accuracy, of our proposed methods is provided. A numerical integration application is performed to demonstrate the use of algorithms. Numerical and graphical results are implemented by MAPLE.

Application of modern computer algebra systems in food formulations and development: A case study

2017 ◽  
Vol 64 ◽  
pp. 48-59 ◽  
Author(s):  
Olga Musina ◽  
Predrag Putnik ◽  
Mohamed Koubaa ◽  
Francisco J. Barba ◽  
Ralf Greiner ◽  
...  
Keyword(s):  
Computer Algebra ◽  
Computer Algebra Systems ◽  
Modern Computer

scholarly journals A novel-iterative simulation method for performance analysis of non-coherent FSK/ASK systems over Rice/Rayleigh channels using the wolfram language

2016 ◽  
Vol 13 (2) ◽  
pp. 157-174 ◽  
Author(s):  
Vladimir Mladenovic ◽  
Danijela Milosevic
Keyword(s):  
Closed Form ◽  
Communication Systems ◽  
Rayleigh Fading ◽  
Simulation Method ◽  
Rician Fading ◽  
New Approach ◽  
Closed Form Solutions ◽  
Numerical Tools ◽  
The Impact ◽  
Insight Into

In this paper, a new approach in solving and analysing the performances of the digital telecommunication non-coherent FSK/ASK system in the presence of noise is derived, by using a computer algebra system. So far, most previous solutions cannot be obtained in closed form, which can be a problem for detailed analysis of complex communication systems. In this case, there is no insight into the influence of certain parameters on the performance of the system. The analysis, modelling and design can be time-consuming. One of the main reasons is that these solutions are obtained by utilising traditional numerical tools in the shape of closed-form expressions. Our results were obtained in closed-form solutions. They are resolved by the introduction of an iteration-based simulation method. The Wolfram language is used for describing applied symbolic tools, and SchematicSolver application package has been used for designing. In a new way, the probability density function and the impact of the newly introduced parameter of iteration are performed when errors are calculated. Analyses of the new method are applied to several scenarios: without fading, in the presence of Rayleigh fading, Rician fading, and in cases when the signals are correlated and uncorrelated.

Some New Results on the Kinematics of 3R Serial Robots Using Nested Determinants

2017 ◽  
Author(s):  
Federico Thomas ◽  
M. Alba Perez-Gracia
Keyword(s):  
Computer Algebra ◽  
Inverse Kinematics ◽  
High Order ◽  
Computer Algebra Systems ◽  
Quadratic Polynomials ◽  
Serial Robots ◽  
Singularity Locus

It has been recently shown that the singularity locus of a 3R robot, and in particular its nodes and cusps, can be algebraically characterized in terms of nested determinants. This neat and structured formulation contrasts with the huge and often meaningless formulas generated using computer algebra systems. In this paper we explore further this kind of formulation. We present two new results which we think are of interest by themselves. First, it is shown how Chrystal’s method, used to obtain the resultant of two quadratic polynomials, can be formulated as nested determinants. Second, it is also shown how the coefficients of the harmonic conic of two given conics, can also be expressed in the same form. These results lead to new formulations for the inverse kinematics of 3R robots, their singularity loci, their nodes, and some of their high-order singularities.

scholarly journals Reconfigurable Intelligent Surface assisted Two–Way Communications: Performance Analysis and Optimization

2020 ◽  
Author(s):  
Saman Atapattu ◽  
Rongfei Fan ◽  
Prathapasinghe Dharmawansa ◽  
Gongpu Wang ◽  
JAMIE EVANS ◽  
...  
Keyword(s):  
Closed Form ◽  
Fading Channels ◽  
Spectral Efficiency ◽  
Rayleigh Fading ◽  
Performance Metrics ◽  
Optimal Solution ◽  
Single Element ◽  
Element Analysis ◽  
Closed Form Solutions ◽  
Semidefinite Programming Problem

In this paper, we investigate the two-way communication between two users assisted by a re-configurable intelligent surface (RIS). The scheme that two users communicate simultaneously in the same time slot over Rayleigh fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at $\left(\log(\rho)/\rho\right)^L$ rate where $L$ is the number of elements, whereas the spectral efficiency increases at $\log(\rho)$ rate at large average SINR $\rho$. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solutions are derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter is relaxed to be a semidefinite programming problem, whose optimal solution is achievable and serves as a sub-optimal solution.

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