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Hong Zheng - One of the best experts on this subject based on the ideXlab platform.
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A partition-of-unity based three-node triangular element with continuous nodal stress using radial-Polynomial Basis functions
Science China Technological Sciences, 2017Co-Authors: Yongtao Yang, Hong ZhengAbstract:A partition-of-unity (PU) based “FE-Meshfree” three-node triangular element (Trig3-RPIM) was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element (Trig3) and radial-Polynomial Basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom (DOFs), the nodal stress field through Trig3-RPIM element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.
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Application of the ‘FE-Meshfree’ QUAD4 with continuous nodal stress using radial-Polynomial Basis functions for vibration and geometric nonlinear analyses
Engineering Analysis With Boundary Elements, 2017Co-Authors: Yongtao Yang, Hong ZhengAbstract:Abstract A hybrid ‘FE-Meshfree’ four-node quadrilateral element with continuous nodal stress using radial-Polynomial Basis functions (Quad4-RPIMcns), was recently proposed for static analysis. The Quad4-RPIMcns element can be considered as a development of the previous partition-of-unity (PU) based ‘FE-Meshfree’ QUAD4 element (Quad4-RPIM) which uses FE shape functions to construct the PU and radial-Polynomial Basis functions to construct the local approximation (LA), so as to synergize the individual strengths of finite element and meshfree methods. As a result, high order global approximations in Quad4-RPIMcns element could be easily constructed without adding extra nodes and DOFs, thereby achieving high accuracy and convergence rate. In this paper, the element is further applied to conduct free vibration, forced vibration and geometric nonlinear analyses of two-dimensional solids. Several numerical test problems are solved and the performance of the element is compared with that of the three-node triangular element (Trig3) and four-node isoparametric quadrilateral element (Quad4). Numerical results show that Quad4-RPIMcns element has higher tolerance to mesh distortion and gives more accurate solution as compared to Trig3 and Quad4 elements.
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A partition-of-unity based ‘FE-Meshfree’ triangular element with radial-Polynomial Basis functions for static and free vibration analysis
Engineering Analysis With Boundary Elements, 2016Co-Authors: Yongtao Yang, Dongdong Xu, Hong ZhengAbstract:Abstract A new ‘FE-Meshfree’ three-node triangular element (Trig3-RPIM) is developed based on the partition of unity (PU) concept. The Trig3-RPIM element employs the shape function of classical three-node triangular element (Trig3) to construct the PU and the radial-Polynomial Basis function which is free from the possible singularity of the moment matrix to construct the nodal approximation. The Trig3-RPIM element synergizes the individual strengths of finite element method and meshfree method. Moreover, it is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several linear, nonlinear and free vibration test problems are solved and the performance of the element is compared with those of the well-known three-node triangular element (Trig3) and four-node iso-parametric quadrilateral element (Quad4). The results show that, for regular meshes, the performance of the element is superior to those of Trig3 and Quad4 elements. For distorted meshes, the present element has better mesh-distortion tolerance than Trig3 and Quad4 elements.
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a hybrid fe meshless quad4 with continuous nodal stress using radial Polynomial Basis functions
Engineering Analysis With Boundary Elements, 2015Co-Authors: Yongtao Yang, Hong ZhengAbstract:Abstract In the present work, a novel hybrid FE-Meshless quadrilateral element with continuous nodal stress is developed using radial-Polynomial Basis functions, named as Quad4-RPIMcns. Quad4-RPIMcns can be regarded as the development of the previous FE-Meshless quadrilateral element with radial-Polynomial Basis functions (Quad4-RPIM) and quadrilateral element with continuous nodal stress (Quad4-CNS). Similar to Quad4-RPIM, radial-Polynomial Basis functions are used to construct nodal approximations of Quad4-RPIMcns in the context of partition of unity, which avoids the possible singularity problem of constructing nodal approximations. The derivative of Quad4-RPIMcns shape function is continuous at nodes. Therefore, nodal stress can be obtained without any extra operation. Quad4-RPIMcns possesses Kronecker-delta property which is a very important property to impose essential boundary conditions directly as in the FEM. The numerical tests in this paper demonstrate that Quad4-RPIMcns gives better accuracy and higher convergence rate as compared to four-node iso-parametric quadrilateral element (Quad4). Additionally, Quad4-RPIMcns seems to have higher tolerance to mesh distortion than Quad4.
Yongtao Yang - One of the best experts on this subject based on the ideXlab platform.
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A partition-of-unity based three-node triangular element with continuous nodal stress using radial-Polynomial Basis functions
Science China Technological Sciences, 2017Co-Authors: Yongtao Yang, Hong ZhengAbstract:A partition-of-unity (PU) based “FE-Meshfree” three-node triangular element (Trig3-RPIM) was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element (Trig3) and radial-Polynomial Basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom (DOFs), the nodal stress field through Trig3-RPIM element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.
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Application of the ‘FE-Meshfree’ QUAD4 with continuous nodal stress using radial-Polynomial Basis functions for vibration and geometric nonlinear analyses
Engineering Analysis With Boundary Elements, 2017Co-Authors: Yongtao Yang, Hong ZhengAbstract:Abstract A hybrid ‘FE-Meshfree’ four-node quadrilateral element with continuous nodal stress using radial-Polynomial Basis functions (Quad4-RPIMcns), was recently proposed for static analysis. The Quad4-RPIMcns element can be considered as a development of the previous partition-of-unity (PU) based ‘FE-Meshfree’ QUAD4 element (Quad4-RPIM) which uses FE shape functions to construct the PU and radial-Polynomial Basis functions to construct the local approximation (LA), so as to synergize the individual strengths of finite element and meshfree methods. As a result, high order global approximations in Quad4-RPIMcns element could be easily constructed without adding extra nodes and DOFs, thereby achieving high accuracy and convergence rate. In this paper, the element is further applied to conduct free vibration, forced vibration and geometric nonlinear analyses of two-dimensional solids. Several numerical test problems are solved and the performance of the element is compared with that of the three-node triangular element (Trig3) and four-node isoparametric quadrilateral element (Quad4). Numerical results show that Quad4-RPIMcns element has higher tolerance to mesh distortion and gives more accurate solution as compared to Trig3 and Quad4 elements.
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A partition-of-unity based ‘FE-Meshfree’ triangular element with radial-Polynomial Basis functions for static and free vibration analysis
Engineering Analysis With Boundary Elements, 2016Co-Authors: Yongtao Yang, Dongdong Xu, Hong ZhengAbstract:Abstract A new ‘FE-Meshfree’ three-node triangular element (Trig3-RPIM) is developed based on the partition of unity (PU) concept. The Trig3-RPIM element employs the shape function of classical three-node triangular element (Trig3) to construct the PU and the radial-Polynomial Basis function which is free from the possible singularity of the moment matrix to construct the nodal approximation. The Trig3-RPIM element synergizes the individual strengths of finite element method and meshfree method. Moreover, it is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several linear, nonlinear and free vibration test problems are solved and the performance of the element is compared with those of the well-known three-node triangular element (Trig3) and four-node iso-parametric quadrilateral element (Quad4). The results show that, for regular meshes, the performance of the element is superior to those of Trig3 and Quad4 elements. For distorted meshes, the present element has better mesh-distortion tolerance than Trig3 and Quad4 elements.
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a hybrid fe meshless quad4 with continuous nodal stress using radial Polynomial Basis functions
Engineering Analysis With Boundary Elements, 2015Co-Authors: Yongtao Yang, Hong ZhengAbstract:Abstract In the present work, a novel hybrid FE-Meshless quadrilateral element with continuous nodal stress is developed using radial-Polynomial Basis functions, named as Quad4-RPIMcns. Quad4-RPIMcns can be regarded as the development of the previous FE-Meshless quadrilateral element with radial-Polynomial Basis functions (Quad4-RPIM) and quadrilateral element with continuous nodal stress (Quad4-CNS). Similar to Quad4-RPIM, radial-Polynomial Basis functions are used to construct nodal approximations of Quad4-RPIMcns in the context of partition of unity, which avoids the possible singularity problem of constructing nodal approximations. The derivative of Quad4-RPIMcns shape function is continuous at nodes. Therefore, nodal stress can be obtained without any extra operation. Quad4-RPIMcns possesses Kronecker-delta property which is a very important property to impose essential boundary conditions directly as in the FEM. The numerical tests in this paper demonstrate that Quad4-RPIMcns gives better accuracy and higher convergence rate as compared to four-node iso-parametric quadrilateral element (Quad4). Additionally, Quad4-RPIMcns seems to have higher tolerance to mesh distortion than Quad4.
Haining Fan - One of the best experts on this subject based on the ideXlab platform.
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low space complexity crt based bit parallel gf 2n Polynomial Basis multipliers for irreducible trinomials
Integration, 2017Co-Authors: Jiajun Zhang, Haining FanAbstract:Abstract This paper presents new space complexity records for the fastest parallel GF ( 2 n ) multipliers for about 22% values of n such that a degree- n irreducible trinomial f = u n + u k + 1 exists over GF ( 2 ) . By selecting the largest possible value of k ∈ ( n / 2 , 2 n / 3 ] , we further reduce the space complexities of the Chinese remainder theorem (CRT)-based hybrid Polynomial Basis multipliers. Our experimental results show that among the 539 values of n ∈ [ 5 , 999 ] such that f is irreducible for some k ∈ [ 2 , n − 2 ] , there are 317 values of n such that k ∈ ( n / 2 , 2 n / 3 ] . For these irreducible trinomials, the space complexities of the CRT-based hybrid multipliers are reduced by 14.3% on average. As a comparison, the previous CRT-based multipliers considered the case k ∈ [ 2 , n / 2 ] , and the improvement rate is 8.4% on average for only 290 values of n among these 539 values of n .
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a chinese remainder theorem approach to bit parallel gf 2 n Polynomial Basis multipliers for irreducible trinomials
IEEE Transactions on Computers, 2016Co-Authors: Haining FanAbstract:We show that the step “modulo the degree- $n$ field generating irreducible Polynomial” in the classical definition of the $GF(2^{n})$ multiplication operation can be avoided. This leads to an alternative representation of the finite field multiplication operation. Combining this representation and the Chinese Remainder Theorem, we design bit-parallel $GF(2^{n})$ multipliers for irreducible trinomials $u^n+u^k+1$ on $GF(2)$ where $1 . For some values of $n$ , our architectures have the same time complexity as the fastest bit-parallel multipliers—the quadratic multipliers, but their space complexities are reduced. Take the special irreducible trinomial $u^{2k}+u^k+1$ for example, the space complexity of the proposed design is reduced by about $1/8$ , while the time complexity matches the best result. Our experimental results show that among the 539 values of $n$ such that $4 and $x^n+x^k+1$ is irreducible over $GF(2)$ for some $k$ in the range $1 , the proposed multipliers beat the current fastest parallel multipliers for 290 values of $n$ when $(n-1)/3 \le k \le n/2$ : they have the same time complexity, but the space complexities are reduced by $8.4$ percent on average.
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schmi gf 2 n shifted Polynomial Basis multipliers based on subquadratic toeplitz matrix vector product approach for all irreducible pentanomials
IEEE Transactions on Computers, 2015Co-Authors: Jiangtao Han, Haining FanAbstract:Besides Karatsuba’s algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design $GF(2^n)$ subquadratic multipliers. However, when $GF(2^n)$ elements are represented using a Polynomial Basis or its generalization—shifted Polynomial Basis—this approach is currently appliable only to fields $GF(2^n)$ generated by an irreducible trinomial or a special type of irreducible pentanomials, but not to a general irreducible pentanomial. The reason is that no transformation matrix, which transforms the Mastrovito matrix into a Toeplitz matrix, has been found. In this article, we propose such a transformation matrix and its inverse matrix for an arbitrary irreducible pentanomial.
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gf 2 n bit parallel squarer using generalised Polynomial Basis for new class of irreducible pentanomials
Electronics Letters, 2014Co-Authors: Xi Xiong, Haining FanAbstract:Explicit formulae and complexities of bit-parallel GF(2 n ) squarers for a new class of irreducible pentanomials x n + x n-1 + x k + x + 1, where n is odd and 1 <; k <; ( n - 1)/2 are presented. The squarer is based on the generalised Polynomial Basis of GF(2 n ). Its gate delay matches the best results, whereas its XOR gate complexity is n + 1, which is only about two thirds of the current best results.
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fast bit parallel shifted Polynomial Basis multipliers in gf 2 n
IEEE Transactions on Circuits and Systems, 2006Co-Authors: Haining Fan, Masud HasanAbstract:A new nonpipelined bit-parallel-shifted Polynomial Basis multiplier for GF(2n) is presented. For some irreducible trinomials, the space complexity of the multiplier matches the best results available in the literature, and its gate delay is equal to T A+lceillog2nrceilTX, where TA and TX are the delay of one two-input and and xor gates, respectively. To the best of our knowledge, this is the first time that the gate delay bound TA+lceillog2nrceilTX is reached. For some irreducible pentanomials, its gate delay is equal to TA +(1+lceillog2nrceil)TX. NIST has recommended five binary fields for the elliptic curve digital signature algorithm applications: GF(2163), GF(2233), GF(2 283), GF(2409), and GF(2571), but no irreducible trinomials exist for three degrees, viz., 163, 283 and 571. For the three corresponding binary fields, we show that the gate delay of the proposed multiplier is TA+(1+lceillog2nrceil)TX. This result outperforms the previously known results
Arash Reyhanimasoleh - One of the best experts on this subject based on the ideXlab platform.
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digit level semi systolic and systolic structures for the shifted Polynomial Basis multiplication over binary extension fields
IEEE Transactions on Very Large Scale Integration Systems, 2011Co-Authors: Arash Hariri, Arash ReyhanimasolehAbstract:Finite field multiplication is one of the most important operations in the finite field arithmetic. In this paper, we study semi-systolic and systolic implementations of the shifted Polynomial Basis multiplication and propose low time complexity semi-systolic and systolic array structures. We show that our proposed semi-systolic multiplier is faster than its existing counterparts available in the literature. Our application-specified integrated circuit (ASIC) implementation of the proposed semi-systolic multiplier demonstrates that reduction in time complexity is achieved without imposing hardware overhead. Furthermore, our proposed systolic array shifted Polynomial Basis (SPB) multiplier has a low time complexity for general irreducible Polynomials.
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a lightweight concurrent fault detection scheme for the aes s boxes using normal Basis
Cryptographic Hardware and Embedded Systems, 2008Co-Authors: Mehran Mozaffarikermani, Arash ReyhanimasolehAbstract:The use of an appropriate fault detection scheme for hardware implementation of the Advanced Encryption Standard (AES) makes the standard robust to the internal defects and fault attacks. To minimize the overhead cost of the fault detection AES structure, we present a lightweight concurrent fault detection scheme for the composite field realization of the S-box using normal Basis. The structure of the S-box is divided into blocks and the predicted parities of these blocks are obtained. Through an exhaustive search among all available composite fields and transformation matrices that map the Polynomial Basis representation in binary field to the normal Basis representation in composite field, we have found the optimum solution for the least overhead S-box and its parity predictions. Finally, using FPGA implementations, the complexities of the proposed schemes are compared to those of the previously reported ones. It is shown that the FPGA implementations of the S-box using normal Basis representation in composite fields outperform the traditional ones using Polynomial Basis for both with and without fault detection capability.
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digit serial structures for the shifted Polynomial Basis multiplication over binary extension fields
International conference on Arithmetic of finite fields, 2008Co-Authors: Arash Hariri, Arash ReyhanimasolehAbstract:Finite field multiplication is one of the most important operations in the finite field arithmetic. Recently, a variation of the Polynomial Basis, which is known as the shifted Polynomial Basis, has been introduced. Current research shows that this new Basis provides better performance in designing bit-parallel and subquadratic space complexity multipliers over binary extension fields. In this paper, we study digit-serial multiplication algorithms using the shifted Polynomial Basis. They include a Most Significant Digit (MSD)-first digit-serial multiplication algorithm and a hybrid digit-serial multiplication algorithm, which includes parallel computations. Then, we explain the hardware architectures of the proposed algorithms and compare them to their existing counterparts. We show that our MSD-first digit-serial shifted Polynomial Basis multiplier has the same complexity of the Least Significant Digit (LSD)-first Polynomial Basis multiplier. Also, we present the results for the hybrid digit-serial multiplier which offers almost the half of the latency of the best known digit-serial Polynomial Basis multipliers.
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low complexity bit parallel architectures for Polynomial Basis multiplication over gf 2m
IEEE Transactions on Computers, 2004Co-Authors: Arash Reyhanimasoleh, M A HasanAbstract:Representing the field elements with respect to the Polynomial (or standard) Basis, we consider bit parallel architectures for multiplication over the finite field GF(2m). In this effect, first we derive a new formulation for Polynomial Basis multiplication in terms of the reduction matrix Q. The main advantage of this new formulation is that it can be used with any field defining irreducible Polynomial. Using this formulation, we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix Q. To the best of our knowledge, this is the first time that these complexities are given in terms of Q. Unlike most other articles on bit parallel finite field multipliers, here we also consider the number of signals to be routed in hardware implementation and we show that, compared to the well-known Mastrovito's multiplier, the proposed architecture has fewer routed signals. The proposed generalized architecture is further optimized for three special types of Polynomials, namely, equally spaced Polynomials, trinomials, and pentanomials. We have obtained explicit formulas and complexities of the multipliers for these three special irreducible Polynomials. This makes it very easy for a designer to implement the proposed multipliers using hardware description languages like VHDL and Verilog with minimum knowledge of finite field arithmetic.
S Rajendran - One of the best experts on this subject based on the ideXlab platform.
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A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-Polynomial Basis functions for static analyses
Computer Methods in Applied Mechanics and Engineering, 2011Co-Authors: J.p. Xu, S RajendranAbstract:Abstract Recently, a partition-of-unity (PU) based finite element called ‘FE-Meshfree’ QUAD4 element has been published. That element employed Polynomial Basis functions for the local approximation (LA). In the present paper, a new FE-Meshfree QUAD4 element employing hybrid radial-Polynomial Basis functions for the LA is proposed. An advantage of radial-Polynomial Basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of Polynomial Basis functions. Another advantage is the improved accuracy of finite element solution. The new element has been applied to several linear and nonlinear test problems. The results demonstrate that the new element gives much better performance as compared to the previous FE-Meshfree QUAD4 element with pure Polynomial Basis. Even with a lower order Basis, viz., with just three or four Polynomial terms, the present element is capable of giving more accurate solution than the previous element which employs a six-term Polynomial Basis. The new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements. Moreover, the linear dependence problem, otherwise associated with many of the partition-of-unity (PU) based elements, is completely eliminated from the present element.