Mathematically, why does mass matrix / load vector lumping work?What does symmetrize mean? (imposing multifreedom constraints to stiffness matrix)Why does FEM usually formulate the problems in reference configuration?How to formulate lumped mass matrix in FEMHow does the animation work in eigenvalue problem of FEMIn FEM, why is the stiffness matrix positive definite?How is the mass matrix formed in finite element methods?Effects of Lumping Mass MatrixMass Lumping in case of Dirichlet boundary conditionsComposite Laminate Mass Matrix HelpCan a second-order ODE be “inconsistent” with its boundary conditions?
SOQL Not Recognizing Field?
What is the highest possible permanent AC at character creation?
Logarithm of exponential
C++ Arduino IDE receiving garbled `char` from function
Mathematically, why does mass matrix / load vector lumping work?
What is the actual quality of machine translations?
Should I give professor gift at the beginning of my PhD?
Recommended tools for graphs and charts
Were Alexander the Great and Hephaestion lovers?
Using "subway" as name for London Underground?
What makes Ada the language of choice for the ISS's safety-critical systems?
Is open-sourcing the code of a webapp not recommended?
How to deal with apathetic co-worker?
Do simulator games use a realistic trajectory to get into orbit?
Medieval flying castle propulsion
Should an arbiter claim draw at a K+R vs K+R endgame?
Did Milano or Benatar approve or comment on their namesake MCU ships?
Overlapping String-Blocks
Should I avoid hard-packed crusher dust trails with my hybrid?
Would the US government be able to hold control if all electronics were disabled for an indefinite amount of time?
How Often Do Health Insurance Providers Drop Coverage?
What ways have you found to get edits from non-LaTeX users?
What can I, as a user, do about offensive reviews in App Store?
What's up with this leaf?
Mathematically, why does mass matrix / load vector lumping work?
What does symmetrize mean? (imposing multifreedom constraints to stiffness matrix)Why does FEM usually formulate the problems in reference configuration?How to formulate lumped mass matrix in FEMHow does the animation work in eigenvalue problem of FEMIn FEM, why is the stiffness matrix positive definite?How is the mass matrix formed in finite element methods?Effects of Lumping Mass MatrixMass Lumping in case of Dirichlet boundary conditionsComposite Laminate Mass Matrix HelpCan a second-order ODE be “inconsistent” with its boundary conditions?
$begingroup$
I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented code where the load vector is assembled in a lumped fashion rather than an FEM-consistent fashion. But I've never looked into why we are allowed to do this in the first place.
What is the intuition behind lumping that allows one to apply it to mass and load vectors? What is the mathematical justification for it? In what situations is lumping not allowed / not a good approximation for mass and load vectors?
finite-element assembly
asked 8 hours ago
Paul♦Paul
7,369639110
$endgroup$
add a comment |
$begingroup$
I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented code where the load vector is assembled in a lumped fashion rather than an FEM-consistent fashion. But I've never looked into why we are allowed to do this in the first place.
What is the intuition behind lumping that allows one to apply it to mass and load vectors? What is the mathematical justification for it? In what situations is lumping not allowed / not a good approximation for mass and load vectors?
finite-element assembly
asked 8 hours ago
Paul♦Paul
7,369639110
$endgroup$
add a comment |
$begingroup$
I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented code where the load vector is assembled in a lumped fashion rather than an FEM-consistent fashion. But I've never looked into why we are allowed to do this in the first place.
What is the intuition behind lumping that allows one to apply it to mass and load vectors? What is the mathematical justification for it? In what situations is lumping not allowed / not a good approximation for mass and load vectors?
finite-element assembly
asked 8 hours ago
Paul♦Paul
7,369639110
$endgroup$
I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented code where the load vector is assembled in a lumped fashion rather than an FEM-consistent fashion. But I've never looked into why we are allowed to do this in the first place.
What is the intuition behind lumping that allows one to apply it to mass and load vectors? What is the mathematical justification for it? In what situations is lumping not allowed / not a good approximation for mass and load vectors?
finite-element assembly
finite-element assembly
asked 8 hours ago
Paul♦Paul
7,369639110
asked 8 hours ago
Paul♦Paul
7,369639110
edited 6 hours ago
Paul
asked 8 hours ago
Paul♦Paul
7,369639110
asked 8 hours ago
Paul♦Paul
7,369639110
asked 8 hours ago
Paul♦Paul
7,369639110
7,369639110
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature formulas one could choose, and one often chooses them in a way so that (i) the error introduced by quadrature is of the same order as that due to discretization, or at least not substantially worse, and (ii) the matrix has certain properties that turn out to be convenient.
Mass lumping is an example of this working: If one chooses a particular quadrature formula (namely, the one with quadrature points located at the interpolation points of the finite element), then the resulting mass matrix happens to be diagonal. That's quite convenient for the computational implementation, and the reason why people use these quadrature formulas. It's also the reason why it "works": This particular choice of quadrature formula still has reasonably high order.
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
$endgroup$
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
1
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "363"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fscicomp.stackexchange.com%2fquestions%2f32805%2fmathematically-why-does-mass-matrix-load-vector-lumping-work%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature formulas one could choose, and one often chooses them in a way so that (i) the error introduced by quadrature is of the same order as that due to discretization, or at least not substantially worse, and (ii) the matrix has certain properties that turn out to be convenient.
Mass lumping is an example of this working: If one chooses a particular quadrature formula (namely, the one with quadrature points located at the interpolation points of the finite element), then the resulting mass matrix happens to be diagonal. That's quite convenient for the computational implementation, and the reason why people use these quadrature formulas. It's also the reason why it "works": This particular choice of quadrature formula still has reasonably high order.
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
$endgroup$
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
1
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
add a comment |
$begingroup$
In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature formulas one could choose, and one often chooses them in a way so that (i) the error introduced by quadrature is of the same order as that due to discretization, or at least not substantially worse, and (ii) the matrix has certain properties that turn out to be convenient.
Mass lumping is an example of this working: If one chooses a particular quadrature formula (namely, the one with quadrature points located at the interpolation points of the finite element), then the resulting mass matrix happens to be diagonal. That's quite convenient for the computational implementation, and the reason why people use these quadrature formulas. It's also the reason why it "works": This particular choice of quadrature formula still has reasonably high order.
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
$endgroup$
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
1
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
add a comment |
$begingroup$
In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature formulas one could choose, and one often chooses them in a way so that (i) the error introduced by quadrature is of the same order as that due to discretization, or at least not substantially worse, and (ii) the matrix has certain properties that turn out to be convenient.
Mass lumping is an example of this working: If one chooses a particular quadrature formula (namely, the one with quadrature points located at the interpolation points of the finite element), then the resulting mass matrix happens to be diagonal. That's quite convenient for the computational implementation, and the reason why people use these quadrature formulas. It's also the reason why it "works": This particular choice of quadrature formula still has reasonably high order.
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
$endgroup$
In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature formulas one could choose, and one often chooses them in a way so that (i) the error introduced by quadrature is of the same order as that due to discretization, or at least not substantially worse, and (ii) the matrix has certain properties that turn out to be convenient.
Mass lumping is an example of this working: If one chooses a particular quadrature formula (namely, the one with quadrature points located at the interpolation points of the finite element), then the resulting mass matrix happens to be diagonal. That's quite convenient for the computational implementation, and the reason why people use these quadrature formulas. It's also the reason why it "works": This particular choice of quadrature formula still has reasonably high order.
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
answered 5 hours ago
Wolfgang BangerthWolfgang Bangerth
38.4k3578
38.4k3578
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
1
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
add a comment |
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
1
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
$begingroup$
Awesome answer, as always. I would be also very interested in your opinion on the second part of the question, when lumping is not allowed/bad approximation, if anything comes to mind.
$endgroup$
– Anton Menshov♦
4 hours ago
1
1
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@AntonMenshov: It would seem that it would be difficult (maybe impossible?) to get a good approximation via lumping for higher order elements, since (e.g. diagonal) lumping in that case would to be equivalent to a lower order quadrature applied to higher order polynomials.
$endgroup$
– Paul♦
2 hours ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
$begingroup$
@WolfgangBangerth: I think I understand now. So, it’s like using newton-cotes rules for integration instead of gaussian quadrature. Since each lagrange interpolation functions have unit values at one specific node, migrating the quadrature points to the nodes results in only the diagonal terms becoming nonzero (at least, for linear elements).
$endgroup$
– Paul♦
1 hour ago
add a comment |
Thanks for contributing an answer to Computational Science Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fscicomp.stackexchange.com%2fquestions%2f32805%2fmathematically-why-does-mass-matrix-load-vector-lumping-work%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
