This paper presents a physics-constrained data driven method that variationally embeds measured d... more This paper presents a physics-constrained data driven method that variationally embeds measured data in the modeling and analysis framework. The physics-based model is augmented with sparse but high-fidelity data through a variationally derived loss function. The structure of the loss function is analyzed in the context of variational correction to the modeled response wherein loss function penalizes the difference in the modeled response from the measured data that represents the local response of the system. The Variationally Embedded Measured Data (VEMD) method results in forward simulations that are not only driven by boundary and initial conditions but are also augmented by real measurements taken at only a small number of observation points. In the context of forward simulations, the proposed approach can be seen as inducing inductive biases that exploit the difference between the computed and measured quantities in the parametric space. With the help of a model problem, we sh...
Standard displacement-based finite elements show a locking behavior in the modeling of nearly inc... more Standard displacement-based finite elements show a locking behavior in the modeling of nearly incompressible materials. Similar phenomenon is observed in volume conserving elasto-plasticity. This limitation of the displacement based elements arises because modeling incompressible material behavior adds kinematic constraints to an element, i.e., the volume at the integral points is required to remain constant. The elements that are not able to resolve these constraints suffer from volumetric locking which causes their response to be too stiff and may lead to overestimation of collapse load when applied to geomechanics. In this paper, stabilized finite elements for mixed displacement-pressure formulation that are based on multiscale variational method are developed. The new formulation allows equal low-order interpolations for both displacement and pressure fields and is suitable for application in real engineering applications. The performance of the elements is evaluated by numerica...
Ravi Kumar R. Tumkur∗, Arne J. Pearlstein∗∗, Arif Masud∗∗∗, Oleg V. Gendelman∗∗∗∗, Lawrence A. Be... more Ravi Kumar R. Tumkur∗, Arne J. Pearlstein∗∗, Arif Masud∗∗∗, Oleg V. Gendelman∗∗∗∗, Lawrence A. Bergman∗, Alexander F. Vakakis∗∗ ∗Department of Aerospace Engineering ∗∗Department of Mechanical Science and Engineering ∗∗∗Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA ∗∗∗∗Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 3200, Israel
A stabilized mixed finite element method for shear-rate dependent non-Newtonian fluids: 3D benchm... more A stabilized mixed finite element method for shear-rate dependent non-Newtonian fluids: 3D benchmark problems and application to blood flow in bifurcating arteries
This paper investigates a high performance implementation of an Arbi-trary Lagrangian Eulerian mo... more This paper investigates a high performance implementation of an Arbi-trary Lagrangian Eulerian moving mesh technique on shared memory systems using OpenMP environment. Moving mesh techniques are considered an integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and Fluid Structure Interaction (FSI). The mov-ing mesh technique adopted in this work is based on the notion of nodes relocation, subjected to a certain evolution as well as constraint conditions. A conjugate gradi-ent method augmented with preconditioning is employed for solution of the resulting system of equations. The proposed algorithm, initially, reorders the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh scheme. Numerical simulations are conducted on the multicore AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D problems. ...
Abstract. This paper presents a variational multiscale method for developing stabilized finite el... more Abstract. This paper presents a variational multiscale method for developing stabilized finite element formulations for small strain inelasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting finite element formulation allows ar-bitrary combinations of interpolation functions for the displacement and pressure fields, and thus yields a family of stable and convergent elements. Specifically, equal order interpolations that are easy to implement but violate the celebrated B-B condition, become stable and convergent. A nonlinear constitutive model for the superelastic behavior of shape memory alloys is integrated in the multiscale formulation. Numerical tests of the performance of the elements are presented and representative simulations of the superelastic behavior of shape memory alloys are shown.
A stabilized discontinuous Galerkin method is developed for general hyperelastic materials at fin... more A stabilized discontinuous Galerkin method is developed for general hyperelastic materials at finite strains. Starting from a mixed method incorporating Lagrange multipliers along the interface, the displacement formulation is systematically derived through a variational multiscale approach whereby the numerical fine scales are modeled via edge bubble functions. Analytical expressions that are free from user-defined param-eters arise for the weighted numerical flux and stability tensor. In particular, the specific form taken by these derived quantities naturally accounts for evolving geometric nonlinearity as well as discontinuous mate-rial properties. The method is applicable both to problems containing nonconforming meshes or different element types at specific interfaces and to problems consisting of fully discontinuous numerical approxi-mations. Representative numerical tests involving large strains and rotations are performed to confirm the
We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori ... more We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the ‘‘stability norm’’ are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.
This paper presents a physics-constrained data driven method that variationally embeds measured d... more This paper presents a physics-constrained data driven method that variationally embeds measured data in the modeling and analysis framework. The physics-based model is augmented with sparse but high-fidelity data through a variationally derived loss function. The structure of the loss function is analyzed in the context of variational correction to the modeled response wherein loss function penalizes the difference in the modeled response from the measured data that represents the local response of the system. The Variationally Embedded Measured Data (VEMD) method results in forward simulations that are not only driven by boundary and initial conditions but are also augmented by real measurements taken at only a small number of observation points. In the context of forward simulations, the proposed approach can be seen as inducing inductive biases that exploit the difference between the computed and measured quantities in the parametric space. With the help of a model problem, we sh...
Standard displacement-based finite elements show a locking behavior in the modeling of nearly inc... more Standard displacement-based finite elements show a locking behavior in the modeling of nearly incompressible materials. Similar phenomenon is observed in volume conserving elasto-plasticity. This limitation of the displacement based elements arises because modeling incompressible material behavior adds kinematic constraints to an element, i.e., the volume at the integral points is required to remain constant. The elements that are not able to resolve these constraints suffer from volumetric locking which causes their response to be too stiff and may lead to overestimation of collapse load when applied to geomechanics. In this paper, stabilized finite elements for mixed displacement-pressure formulation that are based on multiscale variational method are developed. The new formulation allows equal low-order interpolations for both displacement and pressure fields and is suitable for application in real engineering applications. The performance of the elements is evaluated by numerica...
Ravi Kumar R. Tumkur∗, Arne J. Pearlstein∗∗, Arif Masud∗∗∗, Oleg V. Gendelman∗∗∗∗, Lawrence A. Be... more Ravi Kumar R. Tumkur∗, Arne J. Pearlstein∗∗, Arif Masud∗∗∗, Oleg V. Gendelman∗∗∗∗, Lawrence A. Bergman∗, Alexander F. Vakakis∗∗ ∗Department of Aerospace Engineering ∗∗Department of Mechanical Science and Engineering ∗∗∗Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA ∗∗∗∗Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 3200, Israel
A stabilized mixed finite element method for shear-rate dependent non-Newtonian fluids: 3D benchm... more A stabilized mixed finite element method for shear-rate dependent non-Newtonian fluids: 3D benchmark problems and application to blood flow in bifurcating arteries
This paper investigates a high performance implementation of an Arbi-trary Lagrangian Eulerian mo... more This paper investigates a high performance implementation of an Arbi-trary Lagrangian Eulerian moving mesh technique on shared memory systems using OpenMP environment. Moving mesh techniques are considered an integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and Fluid Structure Interaction (FSI). The mov-ing mesh technique adopted in this work is based on the notion of nodes relocation, subjected to a certain evolution as well as constraint conditions. A conjugate gradi-ent method augmented with preconditioning is employed for solution of the resulting system of equations. The proposed algorithm, initially, reorders the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh scheme. Numerical simulations are conducted on the multicore AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D problems. ...
Abstract. This paper presents a variational multiscale method for developing stabilized finite el... more Abstract. This paper presents a variational multiscale method for developing stabilized finite element formulations for small strain inelasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting finite element formulation allows ar-bitrary combinations of interpolation functions for the displacement and pressure fields, and thus yields a family of stable and convergent elements. Specifically, equal order interpolations that are easy to implement but violate the celebrated B-B condition, become stable and convergent. A nonlinear constitutive model for the superelastic behavior of shape memory alloys is integrated in the multiscale formulation. Numerical tests of the performance of the elements are presented and representative simulations of the superelastic behavior of shape memory alloys are shown.
A stabilized discontinuous Galerkin method is developed for general hyperelastic materials at fin... more A stabilized discontinuous Galerkin method is developed for general hyperelastic materials at finite strains. Starting from a mixed method incorporating Lagrange multipliers along the interface, the displacement formulation is systematically derived through a variational multiscale approach whereby the numerical fine scales are modeled via edge bubble functions. Analytical expressions that are free from user-defined param-eters arise for the weighted numerical flux and stability tensor. In particular, the specific form taken by these derived quantities naturally accounts for evolving geometric nonlinearity as well as discontinuous mate-rial properties. The method is applicable both to problems containing nonconforming meshes or different element types at specific interfaces and to problems consisting of fully discontinuous numerical approxi-mations. Representative numerical tests involving large strains and rotations are performed to confirm the
We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori ... more We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the ‘‘stability norm’’ are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.
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