A Posteriori Error Estimation for Predictive Models

A Posteriori Error Estimation for

Predictive Models

James R. Stewart [email protected]

Engineering Sciences Center

Advanced Computational Mechanics Architectures Dept (9143) Sandia National Laboratories

Error Estimation Example

•

Linear elastic bar with

g

ra

v

ity

lo

a

d

in

g

•

8-node hexahedral

elements

This is among the

simplest problems to

solve using FEA

Q. Can we estimate

the error

accurately

?

A

. Y

e

s

…

Error Estimation Example

What does it depend on?

•

The type of error

estimator

•

The mesh resolution

(possibly)

•

The engineering

“quantity of interest”

» On the bottom surface

– Average normal stress on the the bottom surface

Rigorous

Definition of Error

• Exact error

is the exact solution to the math problem

is the numerical solution (finite-element, finite-volume, finite-difference, etc.)

Non-Rigorous

Definitions of Error

•

Difference between fine and coarse meshes

•

Difference between higher-order and lower-order

meshes

Definitions of Error (cont.)

Many (most) error estimators work directly on

with the assumption that

is small

or

•

The exact error can be written as

where

•

Remark

More Definitions

•

Quantity of interest

– Functional of solution

– Local engineering output

» Average vertical displacement on a surface

» Displacement at a point

Error

Estimates

vs. Error

Bounds

•

Error

estimates

– Attempt to quantify the error (notice the use of )

– Quality of estimate often given by the effectivity index

•

Error

bounds

– Upper and lower bounds

– Can easily convert bounds into estimates

Remarks

•

For Uncertainty Quantification, error

bounds

are more

useful than error

estimates

•

Error

estimates

provide stopping criteria for adaptive

mesh refinement

•

Converting an estimate into bounds is, in general, not

trivial

Error Estimation Example (cont.)

•

Quantity of interest

– Average vertical displacement

– Case 1: is a side surface

– Case 2: is the bottom surface

Lower and Upper Bounds:

Case 1 (Side Surface)

Valid upper

bound on

Valid lower

bound on

• Bounds are with respect to

• Quality of bounds depends on H

Lower and Upper Bounds:

Case 2 (Bottom Surface)

H

2

• Solution ( ) is exact for all H (note that is nodally exact)

• Error estimate (average of bounds) is exact for all H

1 0.5

0

.

1

8

06

.

2

e

8

06

.

2

e

2

.

06

e

8

8

06

.

2

e

8

06

.

2

e

8

06

.

2

e

1

.

₀

8

06

.

2

e

8

06

.

2

e

8

06

.

2

e

1

.

₀

Classes of Error Estimators

•

Recovery methods

•

Residual methods

Recovery Methods

•

Super-convergent Patch Recovery

(ZZ-SPR or ZZ)

– Zienkiewicz and Zhu

– Post process gradients of FE solution on patches of neighboring elements

– Gives global energy-norm estimates (under stringent assumptions)

– Does not lend itself to rigorous mathematical analysis

•

Polynomial Preserving Recovery

(PPR)

– Zhang, et al.

– Post process nodal values of FE solution on patches of neighboring elements

– Gives global energy-norm estimates

Residual Methods

•

Traditional categorizations

» In reality, a misnomer » Provides error estimates

– Implicit residual methods

» Can provide error estimates and bounds

•

Original ideas of

Babuska and Rheinbolt

[1978-81]

with extensions by

Babuska and Miller

[1984-87]

Explicit

Residual Methods

Error Representation Formula

“Weak” residual estimate:

• Solve dual problem for

• Substitute into (3)

Remarks:

• Dual problem must be solved on either or

Explicit

Residual Methods

“Strong” residual estimate:

where and are stability factors

Remarks:

• (4) implies that the error can be large even if the residuals are small

• The stability factors are properties of the pde

Explicit

Residual Methods

Nonlinear Operators

Explicit

Residual Methods

Sample Applications

• Chalmers (Sweden) Group (Johnson, Eriksson, Estep, Hansbo, et al.)

– Advection-diffusion; General nonlinear parabolic operators

– Mostly global norms (done in early 90’s) • Estep, Larson, and Williams

– Nonlinear reaction-diffusion systems

– Coupled parabolic pde’s and (singular) ode’s in time

– = average value in domain

• Heidelberg Group (Becker and Rannacher; Bangerth) – DWR (Dual Weighted Residual) Method

– Variety of nonlinear fluid and solid mechanics problems

– “Improved” error indicators for mesh adaptivity

– (Time-domain) acoustic wave equation; elastic wave equation • Barth and Larson

– Extension to finite volume methods

Implicit

Residual Methods

• Can give error estimates and bounds

• Simplest form (for error estimates): Recall

• Substitute and rearrange; solve for

Remarks:

• Recall ; Therefore, must be solved in a higher-order subspace

• For efficiency, is solved on the broken space • Element residual method

• Subdomain residual method

Implicit

Residual Methods

The

Broken Space

• Broken space can also consist of (overlapping or non-overlapping) patches

Implicit

Residual Methods

Quantity-of-Interest

Bounds

•

Various derivations have been published (beyond the

scope of this talk…), e.g.,

– Babuska and Strouboulis

– Peraire and Patera

– Oden and Prudhomme

•

Main result (broken space dependence made explicit)

•

Subtract to get

error

bounds

• Requirement: (i.e., set of functions defined on also contains functions defined on )

Implicit

Residual Methods

General Bound Procedure

•

Following presentation of

Peraire and Patera

(the

“Truth”

Error Bounds)

•

Step 0:

Solve FE problem

– Find such that

•

Step 1:

Solve global (linear) dual problem on

Implicit

Residual Methods

General Bound Procedure

•

Step 2:

For each patch in , solve

– Local primal error problem: Find such that

– Local dual error problem: Find such that

where is the jump bilinear form defined by

Implicit

Residual Methods

General Bound Procedure

•

Step 3:

Compute the bounds

“Truth” Error Bounds

Software Design and Implementation

Quasi-statics code

Adagio

SIERRA design hierarchy

Domain

Procedure (time step control) Region A

(single step of physics A)

Mechanics

Mesh and Fields

Region B

(single step of physics B)

Mechanics

Mesh and Fields

Adagio

“Truth” Error Bounds

Software Design and Implementation

•

Observations

– We require solution of (1+2N) auxiliary PDE’s (where N is the number of patches in )

» Each has the same lhs (for linear, self-adjoint operator) » Each has, in general, a different rhs

•

The

SIERRA Framework

helps manage some of this

complexity

Adagio

“Truth” Error Bounds

Software Design and Implementation

Primal “Patch” Region

Fields Fields

Dual “Patch” Region

3. Region copy-subset For each patch in

“Global Dual” Region

Mesh and Fields

1. Region copy-subset

Adagio Procedure Adagio Region

Mechanics

Mesh and Fields

Mechanics

Residual Residual

2. Solve dual problem

4. Create patch mesh

Mesh Mesh

5. Transfer field values 6. Solve local problems 7. Transfer field values

8. Local update inner products and norms 9. Compute bounds

“Truth” Error Bounds in

Adagio

Case 1 (Side Surface)

Valid upper

bound on

Valid lower

bound on

Implicit

Residual Methods

Sample Applications

TICAM Group

(

Babuska and Strouboulis

;

Oden and

•

Variety of linear and nonlinear

elliptic

problems

– Elasticity problems

» Local and average displacements and stress components

– Heat equation with nonlinear and orthotropic materials

» Local temperature

– Burgers’ equation

» Local velocity

– Incompressible Navier-Stokes

» Kinetic energy of flow

– Helmholtz

» Local amplitude

– Eigenvalue problems

Extrapolation Methods

•

Richardson Extrapolation

– Applies in the asymptotic convergence region

– Assume is computed with two mesh sizes, and , where

– Using a known convergence rate

– Can now eliminate to get a very accurate approximation of

1

h

h

h

h

α

H.O.T.

)

(

)

(

ch

u

l

u

l

H.O.T.

)

(

)

(

ch

u

l

u

l

c

H.O.T.

1

)

(

)

(

)

(

r

u

l

r

u

l

u

l

h

r

Extrapolation Methods

•

Richardson Extrapolation, cont.

– If is not known (almost always), then a third solution must be obtained on where 2 3

h

h

α

r

e

e

log

)

/

log(

α

)

(

)

(

l

u

l

u

e

)

(

)

(

l

u

l

u

e

(Difference in successive meshes)

constant

h

h

h

h

r

Extrapolation Methods

Converting

Estimates

into

Bounds

•

Grid Convergence Index

(GCI) (

Roache

)

•

Essentially a

factor of safety

•

For two-grid extrapolation,

•

For three-grid extrapolation,

•

(Approximate) quantity-of-interest bounds:

s

F

)

(

grid]

fine

[

u

l

e

F

GCI

3

F

25

.

1

F

)

(

)

(

u

GCI

l

u

l

Extrapolation Methods

Ideal vs. Reality

•

Ideal (monotonic convergence)

Example Problem

g

•

Explicit transient dynamics (DYNA)

Extrapolation Methods

Ideal vs. Reality

•

Reality

Num Elements

Extrapolation Methods

Ideal vs. Reality

•

Reality (how it was actually handled)

• GCI applied to middle three points (these were monotonic)

• Timestep errors were ignored

Areas of Active Research

•

Extension of bounds to include modeling and

uncertainty errors (

Oden and Prudhomme

, others…)

•

Harder problems

– Nonlinear parabolic problems

– Hyperbolic problems

– Multiphysics problems

– Extreme anisotropic materials

•

Bounds of exact error (

Babuska and Strouboulis

,

Peraire, et al.

)

– “Certificates” of precision

•

Errors due to operator splitting (

Estep

)

Areas of Active Research (cont.)

•

Nearby problems (

Hopkins, Roy

)

•

Discretization procedures

– Finite volume techniques

– Semi-discrete time integration (method of lines)

– Application to shells, other element types

– Stabilized methods

•

Adaptive error control

– Error indicators

Example of “Hard” Problem

Sandia Thermal Battery

•

Battery operation

– and are heated above melting temperature, then cooled

– Need to stay melted 1 hour

•

Problem features

– Highly transient

– Nonlinear materials (temp-dep)

– Nonsmooth data (read from table)

– Highly orthotropic materials

– Nonlinear BC’s (radiation, convection)

Error Estimation

Risk Assessment

Problem Class Sandia Code(s) Risk Research Issues

Elliptic

Adagio

Salinas (Freq dom) Calore (steady) Low Nonlinearities Nonsmoothness Anisotropic matls Parabolic Calore Aria Fuego Premo (subsonic) Medium Time errors Finite volume Turbulence Hyperbolic Presto

Salinas (Time dom) Premo (supersonic) High Explicit, lumped timestepping History-dep vars Multiphysics Calagio Fuego/Calore/Syrinx … Med-High Loose coupling – transfer operators

Current Limitations of

Error Estimation Impact

•

Can we identify anything that is limiting the impact

and potential of

a posteriori

error estimation?

(Answer: yes)

– Priorities (of the code customers and commercial vendors)

– Computational cost of the algorithms

– Complexity of implementation

Customer Priorities

•

Commercial customers (code end-users) and

commercial software vendors

1. Robustness 2. CPU cost

3. Memory cost 4. Accuracy

•

Accuracy (and knowledge of accuracy) is important,

but not most important

•

In the current marketplace, customers are not willing

to pay a high price for error estimation capabilities

Customer Priorities (cont.)

•

ASCI

– V & V program has elevated the importance of error estimation

» Solution verification » Model validation

– ASCI is driving the need to develop techniques for complex engineering problems

•

Potential to impact the commercial sector, and change

the way all engineering design and analysis is done

•

Success is not guaranteed!

Summary

•

Primary classes of error estimators

– Recovery methods

– Residual methods (quantity-of-interest estimates and bounds)

– Extrapolation methods

•

(Quantity-of-Interest) A Posteriori error estimation is

relatively mature mainly for elliptic problems

•

Can provide both error estimates and error bounds

(good for UQ)

•

A Posteriori error estimation proven very effective for

adaptive error control

Remark:

The

optimization community

has experience

Computational Cost of the

Algorithms

•

Extrapolation

estimators

– Many (at least three) solutions required

– All solutions must be sufficiently resolved, which increases the cost

•

Residual-based

estimators

– Global dual problems required (although linear)

» For estimates, must be solved on finer (h or p) mesh

» For bounds, additional local (element or patch) problems must be solved for error in dual solution

– Time dependent problems

» Dual problems are backwards in time!

Remark:

The

optimization community

has experience

Complexity of Implementation

•

Extrapolation

estimators

– Richardson extrapolation: Usually simple to implement

» Issues mainly with ability to obtain mesh refining or coarsening tools

– Usually can be thought of as a post-processing tool

•

Residual-based

estimators

– Generally intrusive to the code

– Must solve additional problems (pde’s)

» Compute residuals, additional right-hand-sides, etc » Must have a way of handling hybrid fluxes

• Equilibration (very complex to implement)

• Bank-Weiser projection (simpler but less accurate)

• Use subdomains or overlapping patches (potentially costlier)

Limitations on Applicability

of the Algorithms

•

Extrapolation

estimators

– Issue: applicability outside asymptotic convergence regime

•

Residual-based

estimators

– More algorithm research needed

» Solving the backwards-in-time dual problem

» Semi-discrete (method-of-lines) discretizations (instead of DG) » Hyperbolic problems

» Multiphysics problems (e.g., operator splitting…) » Etc.