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Kurt Otto - One of the best experts on this subject based on the ideXlab platform.
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Design and usability of a PDE solver framework for curvilinear coordinates
Advances in Engineering Software, 2006Co-Authors: Malin Ljungberg, Kurt Otto, Michael ThunéAbstract:Abstract An object-oriented PDE solver framework is a library of software components for numerical solution of partial differential equations, where each component is an object or a group of objects. Given such a framework, the construction of a particular PDE solver consists in selecting and combining suitable components. The present paper is focused on tengo [Ahlander K, Otto K. Software design for finite difference schemes based on Index Notation. Future Generation Comput Syst 2006;22:102–9], an object-oriented PDE solver framework for finite difference methods on structured grids, using tensor abstractions for convenient representation of numerical operators. Here, the design of tengo is extended to address curvilinear coordinates. These extensions to the tengo object model are the result of applying object-oriented analysis and design combined with feature modeling. The framework was implemented in Fortran 90/95, using standard techniques for emulating object-oriented constructs in that language. The new parts of the framework were assessed with respect to programming effort and execution time. It is shown that the programming effort required for construction and modification of PDE solvers on curvilinear grids is significantly reduced through the introduction of the new framework components. Moreover, for the test case of an underwater acoustics computation, there was no significant difference in execution time between the framework based code and a special purpose Fortran 90 code for the same application.
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Software design for finite difference schemes based on Index Notation
Future Generation Computer Systems, 2003Co-Authors: Krister Åhlander, Kurt OttoAbstract:A formulation of finite difference schemes based on the Index Notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for higher space dimensions, it is claimed that a band tensor formulation better corresponds to the inherent problem structure than does conventional matrix Notation.Tensor algebra is commonly expressed using Index Notation. The standard Index Notation is extended with the notion of Index offsets, thereby allowing the common traversal of band tensor diagonals.The transition from mathematical Index Notation to implementation is presented. It is emphasized that efficient band tensor computations must exploit the particular problem structure, which calls for a combination of general Index Notation software with special-purpose band tensor routines.
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on software support for finite difference schemes based on Index Notation
International Conference on Computational Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract:A formulation of finite difference schemes based on the Index Notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for 3D, it is claimed that Index Notation better corresponds to the inherent problem structure than does conventional matrix Notation. The transition from mathematical Index Notation to implementation is discussed. Software support for Index Notation that obeys the Einstein summation convention has been implemented in the C++ package Ein-Sum. The extension of EinSum to support typical data structures of finite difference schemes is outlined. A combination of general Index Notation software and special-purpose routines for instance for fast transforms is envisioned.
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International Conference on Computational Science (3) - On Software Support for Finite Difference Schemes Based on Index Notation
Lecture Notes in Computer Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract:A formulation of finite difference schemes based on the Index Notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for 3D, it is claimed that Index Notation better corresponds to the inherent problem structure than does conventional matrix Notation. The transition from mathematical Index Notation to implementation is discussed. Software support for Index Notation that obeys the Einstein summation convention has been implemented in the C++ package Ein-Sum. The extension of EinSum to support typical data structures of finite difference schemes is outlined. A combination of general Index Notation software and special-purpose routines for instance for fast transforms is envisioned.
Krister Åhlander - One of the best experts on this subject based on the ideXlab platform.
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Software design for finite difference schemes based on Index Notation
Future Generation Computer Systems, 2003Co-Authors: Krister Åhlander, Kurt OttoAbstract:A formulation of finite difference schemes based on the Index Notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for higher space dimensions, it is claimed that a band tensor formulation better corresponds to the inherent problem structure than does conventional matrix Notation.Tensor algebra is commonly expressed using Index Notation. The standard Index Notation is extended with the notion of Index offsets, thereby allowing the common traversal of band tensor diagonals.The transition from mathematical Index Notation to implementation is presented. It is emphasized that efficient band tensor computations must exploit the particular problem structure, which calls for a combination of general Index Notation software with special-purpose band tensor routines.
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on software support for finite difference schemes based on Index Notation
International Conference on Computational Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract:A formulation of finite difference schemes based on the Index Notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for 3D, it is claimed that Index Notation better corresponds to the inherent problem structure than does conventional matrix Notation. The transition from mathematical Index Notation to implementation is discussed. Software support for Index Notation that obeys the Einstein summation convention has been implemented in the C++ package Ein-Sum. The extension of EinSum to support typical data structures of finite difference schemes is outlined. A combination of general Index Notation software and special-purpose routines for instance for fast transforms is envisioned.
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International Conference on Computational Science (3) - On Software Support for Finite Difference Schemes Based on Index Notation
Lecture Notes in Computer Science, 2002Co-Authors: Krister Åhlander, Kurt OttoAbstract:A formulation of finite difference schemes based on the Index Notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, "tensors". Especially for 3D, it is claimed that Index Notation better corresponds to the inherent problem structure than does conventional matrix Notation. The transition from mathematical Index Notation to implementation is discussed. Software support for Index Notation that obeys the Einstein summation convention has been implemented in the C++ package Ein-Sum. The extension of EinSum to support typical data structures of finite difference schemes is outlined. A combination of general Index Notation software and special-purpose routines for instance for fast transforms is envisioned.
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Einstein summation for multidimensional arrays
Computers & Mathematics with Applications, 2002Co-Authors: Krister ÅhlanderAbstract:Abstract One of the most common data structures, at least in scientific computing, is the multidimensional array. Some numerical algorithms may conveniently be expressed as a generalized matrix multiplication, which computes a multidimensional array from two other multidimensional arrays. By adopting Index Notation with the Einstein summation convention, an elegant tool for expressing generalized matrix multiplications is obtained. Index Notation is the succinct and compact Notation primarily used in tensor calculus. In this paper, we develop computer support for Index Notation as a domain specific language. Grammar and semantics are proposed, yielding an unambiguous interpretation algorithm. An object-oriented implementation of a C++ library that supports Index Notation is described. A key advantage with computer support of Index Notation is that the Notational gap between a mathematical Index Notation algorithm and its implementation in a computer language is avoided. This facilitates program construction as well as program understanding. Program examples that demonstrate the close resemblance between code and the original mathematical formulation are presented.
Satoshi Egi - One of the best experts on this subject based on the ideXlab platform.
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Scalar and Tensor Parameters for Importing the Notation in Differential Geometry into Programming
arXiv: Programming Languages, 2018Co-Authors: Satoshi EgiAbstract:This paper proposes a method for importing tensor Index Notation, including Einstein summation Notation, into programming. This method involves introducing two types of parameters, i.e, scalar and tensor parameters. As an ordinary function, when a tensor parameter obtains a tensor as an argument, the function treats the tensor argument as a whole. In contrast, when a scalar parameter obtains a tensor as an argument, the function is applied to each component of the tensor. This paper shows that introducing these two types of parameters enables us to apply arbitrary functions to tensor arguments using Index Notation without requiring an additional description to enable each function to handle tensors. Furthermore, we show this method can be easily extended to define concisely the operators for differential forms such as the wedge product, exterior derivative, and Hodge star operator. It is achieved by providing users the method for controlling the completion of omitted indices.
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Scalar Functions and Tensor Functions: A Method to Import Tensor Index Notation Including Einstein Summation Notation.
arXiv: Programming Languages, 2017Co-Authors: Satoshi EgiAbstract:In this paper, we import tensor Index Notation including Einstein summation Notation into programming by introducing two kinds of functions, tensor functions and scalar functions. Tensor functions are functions that contract the tensors given as an argument, and scalar functions are the others. As with ordinary functions, when a tensor function obtains a tensor as an argument, the tensor function treats the tensor as it is as a tensor. On the other hand, when a scalar function obtains a tensor as an argument, the scalar function is applied to each component of the tensor. This paper shows that, by introducing these two kinds of functions, Index Notation can be imported into whole programming, that means we can use Index Notation for arbitrary functions, without requiring annoying description to enable each function to handle tensors. This method can be applied to arbitrary programming languages.
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Scalar and Tensor Parameters for Importing Tensor Index Notation including Einstein Summation Notation
arXiv: Programming Languages, 2017Co-Authors: Satoshi EgiAbstract:In this paper, we propose a method for importing tensor Index Notation, including Einstein summation Notation, into functional programming. This method involves introducing two types of parameters, i.e, scalar and tensor parameters, and simplified tensor Index rules that do not handle expressions that are valid only for the Cartesian coordinate system, in which the Index can move up and down freely. An example of such an expression is "c = A_i B_i". As an ordinary function, when a tensor parameter obtains a tensor as an argument, the function treats the tensor argument as a whole. In contrast, when a scalar parameter obtains a tensor as an argument, the function is applied to each component of the tensor. In this paper, we show that introducing these two types of parameters and our simplified Index rules enables us to apply arbitrary user-defined functions to tensor arguments using Index Notation including Einstein summation Notation without requiring an additional description to enable each function to handle tensors.
Saman Amarasinghe - One of the best experts on this subject based on the ideXlab platform.
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CGO - Tensor algebra compilation with workspaces
2019 IEEE ACM International Symposium on Code Generation and Optimization (CGO), 2019Co-Authors: Fredrik Kjolstad, Peter Ahrens, Shoaib Kamil, Saman AmarasingheAbstract:This paper shows how to extend sparse tensor algebra compilers to introduce temporary tensors called workspaces to avoid inefficient sparse data structures accesses. We develop an intermediate representation (IR) for tensor operations called concrete Index Notation that specifies when sub-computations occur and where they are stored. We then describe the workspace transformation in this IR, how to programmatically invoke it, and how the IR is compiled to sparse code. Finally, we show how the transformation can be used to optimize sparse tensor kernels, including sparse matrix multiplication, sparse tensor addition, and the matricized tensor times Khatri-Rao product (MTTKRP). Our results show that the workspace transformation brings the performance of these kernels on par with hand-optimized implementations. For example, we improve the performance of MTTKRP with dense output by up to 35%, and enable generating sparse matrix multiplication and MTTKRP with sparse output, neither of which were supported by prior tensor algebra compilers.
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Sparse Tensor Algebra Optimizations with Workspaces
arXiv: Mathematical Software, 2018Co-Authors: Fredrik Kjolstad, Peter Ahrens, Shoaib Kamil, Saman AmarasingheAbstract:This paper shows how to optimize sparse tensor algebraic expressions by introducing temporary tensors, called workspaces, into the resulting loop nests. We develop a new intermediate language for tensor operations called concrete Index Notation that extends tensor Index Notation. Concrete Index Notation expresses when and where sub-computations occur and what tensor they are stored into. We then describe the workspace optimization in this language, and how to compile it to sparse code by building on prior work in the literature. We demonstrate the importance of the optimization on several important sparse tensor kernels, including sparse matrix-matrix multiplication (SpMM), sparse tensor addition (SpAdd), and the matricized tensor times Khatri-Rao product (MTTKRP) used to factorize tensors. Our results show improvements over prior work on tensor algebra compilation and brings the performance of these kernels on par with state-of-the-art hand-optimized implementations. For example, SpMM was not supported by prior tensor algebra compilers, the performance of MTTKRP on the nell-2 data set improves by 35%, and MTTKRP can for the first time have sparse results.
Erasmo Carrera - One of the best experts on this subject based on the ideXlab platform.
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Micro-, Meso- and Macro-Scale Analysis of Composite Laminates by Unified Theory of Structures
Volume 9: Mechanics of Solids Structures and Fluids; NDE Structural Health Monitoring and Prognosis, 2017Co-Authors: Erasmo Carrera, A. G. De Miguel, Alfonso PaganiAbstract:In the present research, an advanced methodology for the multi-scale analysis of composite structures is proposed. It is based on the Carrera Unified Formulation (CUF), according to which any theory of structures, either 2D plate/shell or 1D beam, can be expressed as a degenerate case of Elasticity by using generalized expansions of the fundamental unknown fields. By using an extensive Index Notation, CUF allows the governing equations of the problem under consideration, and eventually the related finite element arrays, to be stated in terms of fundamental nuclei, which are invariant of the theory approximation order and the analysis scale. In this manner, micro-, meso-, and macro-scale models of composite structures can be formulated with ease and in a unified way, without the need of changing the model paradigms from one scale to the other. The capability of the proposed methodology based on CUF is assessed and the results demonstrate the validity of the approach, whose mathematical formalism is scale-independent, but allows for the simultaneous analysis of composites from global to very local scales in an accurate manner.
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Large-deflection and post-buckling analyses of laminated composite beams by Carrera Unified Formulation
Composite Structures, 2017Co-Authors: Alfonso Pagani, Erasmo CarreraAbstract:Abstract The Carrera Unified Formulation (CUF) was recently extended to deal with the geometric nonlinear analysis of solid cross-section and thin-walled metallic beams (Pagani and Carrera, 2017). The promising results provided enough confidence for exploring the capabilities of that methodology when dealing with large displacements and post-buckling response of composite laminated beams, which is the subject of the present work. Accordingly, by employing CUF, governing nonlinear equations of low- to higher-order beam theories for laminated beams are expressed in this paper as degenerated cases of the three-dimensional elasticity equilibrium via an appropriate Index Notation. In detail, although the provided equations are valid for any one-dimensional structural theory in a unified sense, layer-wise kinematics are employed in this paper through the use of Lagrange polynomial expansions of the primary mechanical variables. The principle of virtual work and a finite element approximation are used to formulate the governing equations in a total Lagrangian manner, whereas a Newton–Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. Several numerical assessments are proposed, including post-buckling of symmetric cross-ply beams and large displacement analysis of asymmetric laminates under flexural and compression loadings.
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Component-wise models for the accurate dynamic and buckling analysis of composite wing structures
Volume 1: Advances in Aerospace Technology, 2016Co-Authors: Erasmo Carrera, Alfonso Pagani, P. H. Cabral, A. Prado, Gustavo H.c. SilvaAbstract:In the present work, a higher-order beam model able to characterize correctly the three-dimensional strain and stress fields with minimum computational efforts is proposed. One-dimensional models are formulated by employing the Carrera Unified Formulation (CUF), according to which the generic 3D displacement field is expressed as the expansion of the primary mechanical variables. In such a way, by employing a recursive Index Notation, the governing equations and the related finite element arrays of arbitrarily refined beam models can be written in a very compact and unified manner. A Component-Wise (CW) approach is developed in this work by using Lagrange polynomials as expanding cross-sectional functions. By using the principle of virtual work and CUF, free vibration and linearized buckling analyses of composite aerospace structures are investigated. The capabilities of the proposed methodology and the advantages over the classical methods and state-of-the-art tools are widely demonstrated by numerical results
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Unified formulation of geometrically nonlinear refined beam theories
Mechanics of Advanced Materials and Structures, 2016Co-Authors: Alfonso Pagani, Erasmo CarreraAbstract:ABSTRACTBy using the Carrera Unified Formulation (CUF) and a total Lagrangian approach, the unified theory of beams including geometrical nonlinearities is introduced in this article. According to CUF, kinematics of one-dimensional structures are formulated by employing an Index Notation and a generalized expansion of the primary variables by arbitrary cross-section functions. Namely, in this work, low- to higher-order beam models with only pure displacement variables are implemented by utilizing Lagrange polynomial expansions of the unknowns on the cross section. The principle of virtual work and a finite element approximation are used to formulate the governing equations, whereas a Newton-Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. By using CUF and three-dimensional Green-Lagrange strain components, the explicit forms of the secant and tangent stiffness matrices of the unified beam element ar...
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Linearized buckling analysis of isotropic and composite beam-columns by Carrera Unified Formulation and dynamic stiffness method
Mechanics of Advanced Materials and Structures, 2016Co-Authors: Erasmo Carrera, Alfonso Pagani, J.r. BanerjeeAbstract:AbstractThis article introduces a one-dimensional (1D) higher-order exact formulation for linearized buckling analysis of beam-columns. The Carrera Unified Formulation (CUF) is utilized and the displacement field is expressed as a generic N-order expansion of the generalized unknown displacement field. The principle of virtual displacements is invoked along with CUF to derive the governing equations and the associated natural boundary conditions in terms of fundamental nuclei, which can be systematically expanded according to N by exploiting an extensive Index Notation. After the closed form solution of the N-order beam-column element is sought, an exact dynamic stiffness (DS) matrix is derived by relating the amplitudes of the loads to those of the responses. The global DS matrix is finally processed through the application of the Wittrick-Williams algorithm to extract the buckling loads of the structure. Isotropic solid and thin-walled cross-section beams as well as laminated composite structures are an...
