Nonlinear Finite Elements For Continua And Structures
01 Jan 2016-
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TL;DR: A new data-driven computational framework is developed to assist in the design and modeling of new material systems and structures and includes the recently developed “self-consistent clustering analysis” method in order to build large databases suitable for machine learning.
385 citations
Cites background from "Nonlinear Finite Elements For Conti..."
...from which the expressions for the second Piola–Kirchhoff stress of each hyperelastic model can be trivially determined, see [96]....
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TL;DR: This review article provides a concise introduction to the basics of the finite cell method, and summarizes recent developments of the technology, with particular emphasis on the research topics in which the author has been actively involved.
Abstract: The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.
271 citations
Cites background or methods from "Nonlinear Finite Elements For Conti..."
...The adaptation d rectly follows the standard finite element formulation, for which details can be found in [20,24,37,80,187]....
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...In this section, we introduce a geometrically nonlinear finite cell formulation in principal directions, based on the logarithmic strain measure [20, 24, 37, 80, 187]....
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...For a computationally efficient implementation of the deformation resetting, the coincidence of linear and geome trically nonlinear elasticity at the deformation and stress free reference configuration is exploited [20, 24, 37, 187]....
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TL;DR: The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the MerrimanBenceOsher (MBO) operator.
Abstract: Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poissons Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the MerrimanBenceOsher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poissons ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well. A multi-material topology optimization approach exploiting the reconciled level-set method.The boundary of each individual material is evolved with a single level set function.Multiple level set functions are reconciled with the MerrimanBenceOsher (MBO) operator.Both 2D and 3D multi-material designs were obtained and used for validate the proposed method.
184 citations
Cites methods from "Nonlinear Finite Elements For Conti..."
...In the current work, this process is done by using the material time derivative approach [66-68]....
[...]
TL;DR: Several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests are reviewed.
Abstract: The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.
155 citations
Cites background or methods from "Nonlinear Finite Elements For Conti..."
...This stress tensor has no physical interpretation, but it is sometimes preferred, due to its symmetry, especially in computational approaches [11,72,93]....
[...]
...When the geometries and boundary conditions of the deforming body are more complex, or application-specific, inverse finite element modelling can be employed [11,72,93,137]....
[...]
TL;DR: In this article, the authors explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis.
142 citations
References
More filters
TL;DR: A new data-driven computational framework is developed to assist in the design and modeling of new material systems and structures and includes the recently developed “self-consistent clustering analysis” method in order to build large databases suitable for machine learning.
385 citations
TL;DR: This review article provides a concise introduction to the basics of the finite cell method, and summarizes recent developments of the technology, with particular emphasis on the research topics in which the author has been actively involved.
Abstract: The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.
271 citations
TL;DR: The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the MerrimanBenceOsher (MBO) operator.
Abstract: Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poissons Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the MerrimanBenceOsher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poissons ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well. A multi-material topology optimization approach exploiting the reconciled level-set method.The boundary of each individual material is evolved with a single level set function.Multiple level set functions are reconciled with the MerrimanBenceOsher (MBO) operator.Both 2D and 3D multi-material designs were obtained and used for validate the proposed method.
184 citations
TL;DR: Several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests are reviewed.
Abstract: The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.
155 citations
TL;DR: In this article, the authors explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis.
142 citations