insikt - Computational Mechanics - # Elastoplastic Finite Element Modeling using Node-Based Virtual Elements

A Node-Based Uniform Strain Virtual Element Method for Modeling Elastoplastic Solids

Q: How can the NVEM formulation be extended to large deformation elastoplasticity with remeshing, and what are the potential advantages compared to standard Gauss integration approaches?

The extension of the Node-Based Uniform Strain Virtual Element Method (NVEM) to large deformation elastoplasticity with remeshing involves leveraging the nodal integration framework that NVEM employs. In this framework, all field variables, including state and history-dependent variables, are associated with the nodes. This characteristic allows for a more straightforward remapping of these variables during the remeshing process, as the connectivity can be reconstructed based on the current set of nodes without the need for complex interpolation or mapping procedures typically required in standard Gauss integration approaches. The potential advantages of using NVEM in large deformation scenarios include: Locking-Free Behavior: NVEM is designed to be locking-free, which is particularly beneficial in nearly incompressible materials where standard Gauss integration methods often suffer from volumetric locking. Simplified Remeshing: The ability to track state variables at the nodes simplifies the remeshing process, as it avoids the need for remapping these variables across the mesh, which can be computationally intensive and error-prone. Flexibility in Mesh Design: The use of virtual elements allows for arbitrary polygonal shapes, which can better conform to complex geometries compared to traditional finite element meshes. Improved Computational Efficiency: By reducing the complexity associated with variable remapping and maintaining accuracy in the representation of the material behavior, NVEM can lead to more efficient computations in large deformation analyses.

Q: What are the limitations of the node-based uniform strain approach, and under what conditions might it perform poorly compared to other locking-free techniques?

The node-based uniform strain approach, while advantageous in many scenarios, does have limitations. Some of these include: Sensitivity to Mesh Quality: The performance of NVEM can be sensitive to the quality of the mesh. Poorly shaped elements or irregular distributions of nodes can lead to inaccuracies in the strain averaging process, potentially affecting the overall solution. Nonlinear Behavior: In cases where the material exhibits highly nonlinear behavior, the averaging of strains at the nodes may not capture localized effects accurately, leading to a loss of detail in the stress distribution. Complex Loading Conditions: Under certain complex loading conditions, such as those involving rapid changes in loading direction or magnitude, the uniform strain assumption may not adequately represent the actual strain state, leading to inaccuracies. Comparison with Other Techniques: While NVEM is locking-free, other locking-free techniques, such as the B-bar method or mixed formulations, may perform better in specific scenarios, particularly where the material behavior is highly sensitive to volumetric changes or where the mesh is not optimally designed for NVEM.

Q: Could the NVEM be adapted to model other types of complex material behavior beyond elastoplasticity, such as damage, fracture, or multi-physics phenomena?

Yes, the NVEM can be adapted to model other types of complex material behavior beyond elastoplasticity, including damage, fracture, and multi-physics phenomena. The flexibility of the virtual element framework allows for the incorporation of various constitutive models and physical phenomena. Some potential adaptations include: Damage Models: NVEM can be extended to include damage mechanics by modifying the constitutive relations to account for degradation of material properties as damage evolves. This can be achieved by integrating damage variables into the nodal integration framework, allowing for a consistent treatment of damage across the mesh. Fracture Mechanics: The method can be adapted to simulate crack propagation by implementing cohesive zone models or phase-field approaches within the NVEM framework. The nodal integration can facilitate the tracking of crack paths and the evolution of stress fields around the crack tips. Multi-Physics Coupling: NVEM can also be extended to handle multi-physics problems, such as thermo-mechanical coupling, where thermal effects influence mechanical behavior. By incorporating additional state variables related to temperature or other physical fields, NVEM can effectively model the interactions between different physical phenomena. Complex Material Behavior: The inherent flexibility of NVEM allows for the modeling of materials exhibiting viscoelasticity, plasticity with strain rate effects, or even phase transformations, by appropriately defining the nodal integration and constitutive laws. In summary, the NVEM's adaptability and its locking-free nature make it a promising candidate for modeling a wide range of complex material behaviors in computational mechanics.

Centrala begrepp

The node-based uniform strain virtual element method (NVEM) enables linearly-precise virtual elements to accurately model elastoplastic solids while avoiding volumetric locking.

Sammanfattning

The paper presents an extension of the recently proposed node-based uniform strain virtual element method (NVEM) to model small strain elastoplastic solids.

Key highlights:

In the NVEM, the strain is averaged at the nodes from the strain of surrounding linearly-precise virtual elements, using a generalization of the node-based uniform strain approach for finite elements.
The nodal strain is then used to sample the weak form at the nodes, leading to a method where all field variables, including state and history-dependent variables, are associated with the nodes.
This node-based formulation can be advantageous for large deformation simulations with remeshing, as it avoids the need to remap state and history variables.
The performance of the NVEM is assessed through various elastoplastic benchmark problems, demonstrating that it is locking-free and enables linearly-precise virtual elements to solve elastoplastic solids accurately.
The paper also presents the elastoplastic constitutive model used and the stabilization approach adopted for the NVEM formulation.

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Statistik

The strain tensor is split into elastic and plastic parts: ε = εe + εp.
The yield function is defined using the von Mises criterion with linear mixed hardening: Φ(σ, β, σy) = √(3/2)||η|| - σy(ε̄p), where η is the relative stress and σy is a function of the accumulated plastic strain ε̄p.
The elastoplastic consistent tangent operator is computed as: Dep = 2G(1 - Δγ/(3G)q̄trial)I + 6G²(Δγ/q̄trial - 1/(3G + Hk + Hi))N̄ N̄T + KmmT.

Citat

"The nodal strain that results from the averaging process is interpreted as the nodal sample of the strain in the nodal integration of the weak form. Consequently, the nodal strain is also used at the constitutive evaluation level."
"As in any nodal integration method, the state and history-dependent variables in the NVEM become associated with the nodes. In practice, this means that in nonlinear computations these variables are tracked only at the nodes."

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A node-based uniform strain virtual element method for elastoplastic solids

by Rodrigo Silv... på arxiv.org 09-18-2024

https://arxiv.org/pdf/2409.10808.pdf

A node-based uniform strain virtual element method for elastoplastic solids

Djupare frågor

How can the NVEM formulation be extended to large deformation elastoplasticity with remeshing, and what are the potential advantages compared to standard Gauss integration approaches?

The extension of the Node-Based Uniform Strain Virtual Element Method (NVEM) to large deformation elastoplasticity with remeshing involves leveraging the nodal integration framework that NVEM employs. In this framework, all field variables, including state and history-dependent variables, are associated with the nodes. This characteristic allows for a more straightforward remapping of these variables during the remeshing process, as the connectivity can be reconstructed based on the current set of nodes without the need for complex interpolation or mapping procedures typically required in standard Gauss integration approaches.
The potential advantages of using NVEM in large deformation scenarios include:

Locking-Free Behavior: NVEM is designed to be locking-free, which is particularly beneficial in nearly incompressible materials where standard Gauss integration methods often suffer from volumetric locking.
Simplified Remeshing: The ability to track state variables at the nodes simplifies the remeshing process, as it avoids the need for remapping these variables across the mesh, which can be computationally intensive and error-prone.
Flexibility in Mesh Design: The use of virtual elements allows for arbitrary polygonal shapes, which can better conform to complex geometries compared to traditional finite element meshes.
Improved Computational Efficiency: By reducing the complexity associated with variable remapping and maintaining accuracy in the representation of the material behavior, NVEM can lead to more efficient computations in large deformation analyses.

What are the limitations of the node-based uniform strain approach, and under what conditions might it perform poorly compared to other locking-free techniques?

The node-based uniform strain approach, while advantageous in many scenarios, does have limitations. Some of these include:

Sensitivity to Mesh Quality: The performance of NVEM can be sensitive to the quality of the mesh. Poorly shaped elements or irregular distributions of nodes can lead to inaccuracies in the strain averaging process, potentially affecting the overall solution.
Nonlinear Behavior: In cases where the material exhibits highly nonlinear behavior, the averaging of strains at the nodes may not capture localized effects accurately, leading to a loss of detail in the stress distribution.
Complex Loading Conditions: Under certain complex loading conditions, such as those involving rapid changes in loading direction or magnitude, the uniform strain assumption may not adequately represent the actual strain state, leading to inaccuracies.
Comparison with Other Techniques: While NVEM is locking-free, other locking-free techniques, such as the B-bar method or mixed formulations, may perform better in specific scenarios, particularly where the material behavior is highly sensitive to volumetric changes or where the mesh is not optimally designed for NVEM.

Could the NVEM be adapted to model other types of complex material behavior beyond elastoplasticity, such as damage, fracture, or multi-physics phenomena?

Yes, the NVEM can be adapted to model other types of complex material behavior beyond elastoplasticity, including damage, fracture, and multi-physics phenomena. The flexibility of the virtual element framework allows for the incorporation of various constitutive models and physical phenomena. Some potential adaptations include:

Damage Models: NVEM can be extended to include damage mechanics by modifying the constitutive relations to account for degradation of material properties as damage evolves. This can be achieved by integrating damage variables into the nodal integration framework, allowing for a consistent treatment of damage across the mesh.

Fracture Mechanics: The method can be adapted to simulate crack propagation by implementing cohesive zone models or phase-field approaches within the NVEM framework. The nodal integration can facilitate the tracking of crack paths and the evolution of stress fields around the crack tips.

Multi-Physics Coupling: NVEM can also be extended to handle multi-physics problems, such as thermo-mechanical coupling, where thermal effects influence mechanical behavior. By incorporating additional state variables related to temperature or other physical fields, NVEM can effectively model the interactions between different physical phenomena.

Complex Material Behavior: The inherent flexibility of NVEM allows for the modeling of materials exhibiting viscoelasticity, plasticity with strain rate effects, or even phase transformations, by appropriately defining the nodal integration and constitutive laws.

In summary, the NVEM's adaptability and its locking-free nature make it a promising candidate for modeling a wide range of complex material behaviors in computational mechanics.