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SCL-7: Wave equation
Prerequisite of this tutorial is SCL-6: Heat equation

Intended learning outcome:
  • practice of time-dependent problem in MoFEM
  • how to formulate and implement code in MoFEM using PETSc time-stepping solver TS
  • TS solver with implicit scheme



Plain program

The plain program for both the implementation of the UDOs (*.spp) and the main program (*.cpp) are as follows

Implementation of the main program (*.cpp)

* \file wave_equation.cpp
* \example wave_equation.cpp
* \brief Solve the time-dependent Wave Equation
\frac{\partial u(\mathbf{x}, t)}{\partial t}-\Delta u(\mathbf{x}, t)
&=f(\mathbf{x}, t) & & \forall \mathbf{x} \in \Omega, t \in(0, T), \\
u(\mathbf{x}, 0) &=u_{0}(\mathbf{x}) & & \forall \mathbf{x} \in \Omega, \\
u(\mathbf{x}, t) &=g(\mathbf{x}, t) & & \forall \mathbf{x} \in \partial \Omega,
t \in(0, T). \end{aligned}
#include <stdlib.h>
#include <cmath>
using namespace MoFEM;
static char help[] = "...\n\n";
template <int DIM> struct ElementsAndOps {};
//! [Define dimension]
constexpr int SPACE_DIM = 2; //< Space dimension of problem, mesh
//! [Define dimension]
constexpr double wave_speed2 = 1;
* @brief Monitor solution
* This functions is called by TS solver at the end of each step. It is used
* to output results to the hard drive.
struct Monitor : public FEMethod {
Monitor(SmartPetscObj<DM> dm, boost::shared_ptr<PostProcEle> post_proc)
: dM(dm), postProc(post_proc){};
MoFEMErrorCode preProcess() { return 0; }
MoFEMErrorCode operator()() { return 0; }
static constexpr int saveEveryNthStep = 1;
if (ts_step % saveEveryNthStep == 0) {
CHKERR postProc->writeFile(
"out_level_" + boost::lexical_cast<std::string>(ts_step) + ".h5m");
boost::shared_ptr<PostProcEle> postProc;
struct WaveEquation {
// Declaration of the main function to run analysis
// Declaration of other main functions called in runProgram()
// Main interfaces
// Object to mark boundary entities for the assembling of domain elements
boost::shared_ptr<std::vector<unsigned char>> boundaryMarker;
WaveEquation::WaveEquation(MoFEM::Interface &m_field) : mField(m_field) {}
CHKERR simple->getOptions();
CHKERR simple->loadFile();
CHKERR simple->addDomainField("U", H1, AINSWORTH_LEGENDRE_BASE, 1);
CHKERR simple->addBoundaryField("U", H1, AINSWORTH_LEGENDRE_BASE, 1);
int order = 3;
CHKERR PetscOptionsGetInt(PETSC_NULL, "", "-order", &order, PETSC_NULL);
CHKERR simple->setFieldOrder("U", order);
CHKERR simple->setUp();
auto integration_rule = [](int o_row, int o_col, int approx_order) {
return 2 * approx_order;
auto *pipeline_mng = mField.getInterface<PipelineManager>();
CHKERR pipeline_mng->setDomainRhsIntegrationRule(integration_rule);
CHKERR pipeline_mng->setDomainLhsIntegrationRule(integration_rule);
CHKERR pipeline_mng->setBoundaryLhsIntegrationRule(integration_rule);
CHKERR pipeline_mng->setBoundaryRhsIntegrationRule(integration_rule);
auto bc_mng = mField.getInterface<BcManager>();
CHKERR bc_mng->pushMarkDOFsOnEntities(simple->getProblemName(), "ESSENTIAL",
"U", 0, 0);
auto &bc_map = bc_mng->getBcMapByBlockName();
boundaryMarker = boost::make_shared<std::vector<char unsigned>>();
for (auto b : bc_map) {
if (std::regex_match(b.first, std::regex("(.*)ESSENTIAL(.*)"))) {
boundaryMarker->resize(b.second->bcMarkers.size(), 0);
for (int i = 0; i != b.second->bcMarkers.size(); ++i) {
(*boundaryMarker)[i] |= b.second->bcMarkers[i];
auto add_domain_base_ops = [&](auto &pipeline) {
auto det_ptr = boost::make_shared<VectorDouble>();
auto jac_ptr = boost::make_shared<MatrixDouble>();
auto inv_jac_ptr = boost::make_shared<MatrixDouble>();
pipeline.push_back(new OpCalculateHOJac<2>(jac_ptr));
pipeline.push_back(new OpInvertMatrix<2>(jac_ptr, det_ptr, inv_jac_ptr));
pipeline.push_back(new OpSetHOInvJacToScalarBases<2>(H1, inv_jac_ptr));
pipeline.push_back(new OpSetHOWeightsOnFace());
auto add_domain_lhs_ops = [&](auto &pipeline) {
pipeline.push_back(new OpSetBc("U", true, boundaryMarker));
pipeline.push_back(new OpDomainGradGrad(
"U", "U", [](double, double, double) -> double { return 1; }));
auto get_c = [this](const double, const double, const double) {
auto pipeline_mng = mField.getInterface<PipelineManager>();
auto &fe_domain_lhs = pipeline_mng->getDomainLhsFE();
return wave_speed2 * fe_domain_lhs->ts_aa;
pipeline.push_back(new OpDomainMass("U", "U", get_c));
pipeline.push_back(new OpUnSetBc("U"));
auto add_domain_rhs_ops = [&](auto &pipeline) {
pipeline.push_back(new OpSetBc("U", true, boundaryMarker));
auto grad_u_at_gauss_pts = boost::make_shared<MatrixDouble>();
auto dot2_u_at_gauss_pts = boost::make_shared<VectorDouble>();
"U", grad_u_at_gauss_pts));
new OpCalculateScalarFieldValuesDotDot("U", dot2_u_at_gauss_pts));
pipeline.push_back(new OpDomainGradTimesVec(
"U", grad_u_at_gauss_pts,
[](double, double, double) -> double { return 1; }));
pipeline.push_back(new OpDomainTimesScalarField(
"U", dot2_u_at_gauss_pts,
[](const double, const double, const double) { return wave_speed2; }));
pipeline.push_back(new OpUnSetBc("U"));
auto add_boundary_base_ops = [&](auto &pipeline) {};
auto add_lhs_base_ops = [&](auto &pipeline) {
pipeline.push_back(new OpSetBc("U", false, boundaryMarker));
pipeline.push_back(new OpBoundaryMass(
"U", "U", [](const double, const double, const double) { return 1; }));
pipeline.push_back(new OpUnSetBc("U"));
auto add_rhs_base_ops = [&](auto &pipeline) {
pipeline.push_back(new OpSetBc("U", false, boundaryMarker));
auto u_at_gauss_pts = boost::make_shared<VectorDouble>();
auto boundary_function = [&](const double x, const double y,
const double z) {
auto pipeline_mng = mField.getInterface<PipelineManager>();
auto &fe_rhs = pipeline_mng->getBoundaryRhsFE();
const double t = fe_rhs->ts_t;
if ((t <= 0.5) && (x < 0.) && (y > -1. / 3) && (y < 1. / 3))
return sin(4 * M_PI * t);
return 0.;
pipeline.push_back(new OpCalculateScalarFieldValues("U", u_at_gauss_pts));
pipeline.push_back(new OpBoundaryTimeScalarField(
"U", u_at_gauss_pts,
[](const double, const double, const double) { return 1; }));
pipeline.push_back(new OpBoundarySource("U", boundary_function));
pipeline.push_back(new OpUnSetBc("U"));
auto pipeline_mng = mField.getInterface<PipelineManager>();
auto *pipeline_mng = mField.getInterface<PipelineManager>();
auto create_post_process_element = [&]() {
auto post_proc_fe = boost::make_shared<PostProcEle>(mField);
auto det_ptr = boost::make_shared<VectorDouble>();
auto jac_ptr = boost::make_shared<MatrixDouble>();
auto inv_jac_ptr = boost::make_shared<MatrixDouble>();
post_proc_fe->getOpPtrVector().push_back(new OpCalculateHOJac<2>(jac_ptr));
new OpInvertMatrix<2>(jac_ptr, det_ptr, inv_jac_ptr));
new OpSetHOInvJacToScalarBases<2>(H1, inv_jac_ptr));
auto u_ptr = boost::make_shared<VectorDouble>();
new OpCalculateScalarFieldValues("U", u_ptr));
new OpPPMap(post_proc_fe->getPostProcMesh(),
{{"U", u_ptr}},
{}, {}, {}
return post_proc_fe;
auto set_time_monitor = [&](auto dm, auto solver) {
boost::shared_ptr<Monitor> monitor_ptr(
new Monitor(dm, create_post_process_element()));
boost::shared_ptr<ForcesAndSourcesCore> null;
CHKERR DMMoFEMTSSetMonitor(dm, solver, simple->getDomainFEName(),
monitor_ptr, null, null);
auto set_fieldsplit_preconditioner = [&](auto solver) {
SNES snes;
CHKERR TSGetSNES(solver, &snes);
KSP ksp;
CHKERR SNESGetKSP(snes, &ksp);
PC pc;
CHKERR KSPGetPC(ksp, &pc);
PetscBool is_pcfs = PETSC_FALSE;
PetscObjectTypeCompare((PetscObject)pc, PCFIELDSPLIT, &is_pcfs);
if (is_pcfs == PETSC_TRUE) {
auto bc_mng = mField.getInterface<BcManager>();
auto name_prb = simple->getProblemName();
auto is_all_bc = bc_mng->getBlockIS(name_prb, "ESSENTIAL", "U", 0, 0);
int is_all_bc_size;
CHKERR ISGetSize(is_all_bc, &is_all_bc_size);
MOFEM_LOG("EXAMPLE", Sev::inform)
<< "Field split block size " << is_all_bc_size;
is_all_bc); // boundary block
auto dm = simple->getDM();
auto D = smartCreateDMVector(dm);
auto ts = pipeline_mng->createTSIM2();
CHKERR TSSetSolution(ts, D);
auto DD = smartVectorDuplicate(D);
CHKERR TS2SetSolution(ts, D, DD);
CHKERR set_time_monitor(dm, ts);
CHKERR TSSetFromOptions(ts);
CHKERR set_fieldsplit_preconditioner(ts);
// Processes to set output results are integrated in solveSystem()
int main(int argc, char *argv[]) {
// Initialisation of MoFEM/PETSc and MOAB data structures
const char param_file[] = "param_file.petsc";
// Add logging channel for example
auto core_log = logging::core::get();
LogManager::createSink(LogManager::getStrmWorld(), "EXAMPLE"));
// Error handling
try {
// Register MoFEM discrete manager in PETSc
DMType dm_name = "DMMOFEM";
// Create MOAB instance
moab::Core mb_instance; // mesh database
moab::Interface &moab = mb_instance; // mesh database interface
// Create MoFEM instance
MoFEM::Core core(moab); // finite element database
MoFEM::Interface &m_field = core; // finite element interface
// Run the main analysis
WaveEquation heat_problem(m_field);
CHKERR heat_problem.runProgram();
// Finish work: cleaning memory, getting statistics, etc.
return 0;
std::string param_file
void simple(double P1[], double P2[], double P3[], double c[], const int N)
Definition: acoustic.cpp:69
static char help[]
int main()
Definition: adol-c_atom.cpp:46
constexpr int SPACE_DIM
ElementsAndOps< SPACE_DIM >::DomainEle DomainEle
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::BiLinearForm< GAUSS >::OpMass< 1, FIELD_DIM > OpDomainMass
ElementsAndOps< SPACE_DIM >::BoundaryEle BoundaryEle
Catch errors.
Definition: definitions.h:372
Ainsworth Cole (Legendre) approx. base .
Definition: definitions.h:60
#define MoFEMFunctionReturnHot(a)
Last executable line of each PETSc function used for error handling. Replaces return()
Definition: definitions.h:447
@ H1
continuous field
Definition: definitions.h:85
#define MoFEMFunctionBegin
First executable line of each MoFEM function, used for error handling. Final line of MoFEM functions ...
Definition: definitions.h:346
#define MoFEMFunctionReturn(a)
Last executable line of each PETSc function used for error handling. Replaces return()
Definition: definitions.h:416
#define CHKERR
Inline error check.
Definition: definitions.h:535
#define MoFEMFunctionBeginHot
First executable line of each MoFEM function, used for error handling. Final line of MoFEM functions ...
Definition: definitions.h:440
ElementsAndOps< SPACE_DIM >::PostProcEle PostProcEle
auto integration_rule
PetscErrorCode DMoFEMMeshToLocalVector(DM dm, Vec l, InsertMode mode, ScatterMode scatter_mode)
set local (or ghosted) vector values on mesh for partition only
Definition: DMMMoFEM.cpp:470
PetscErrorCode DMRegister_MoFEM(const char sname[])
Register MoFEM problem.
Definition: DMMMoFEM.cpp:47
PetscErrorCode DMoFEMLoopFiniteElements(DM dm, const char fe_name[], MoFEM::FEMethod *method, CacheTupleWeakPtr cache_ptr=CacheTupleSharedPtr())
Executes FEMethod for finite elements in DM.
Definition: DMMMoFEM.cpp:533
auto smartCreateDMVector(DM dm)
Get smart vector from DM.
Definition: DMMoFEM.hpp:965
#define MOFEM_LOG(channel, severity)
Definition: LogManager.hpp:301
#define MOFEM_LOG_TAG(channel, tag)
Tag channel.
Definition: LogManager.hpp:332
FTensor::Index< 'i', SPACE_DIM > i
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::BiLinearForm< GAUSS >::OpGradGrad< 1, 1, SPACE_DIM > OpDomainGradGrad
Definition: helmholtz.cpp:27
FormsIntegrators< EdgeEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpSource< 1, 1 > OpBoundarySource
Definition: helmholtz.cpp:33
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpBaseTimesScalar< 1 > OpDomainTimesScalarField
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpGradTimesTensor< 1, 1, SPACE_DIM > OpDomainGradTimesVec
double D
PetscErrorCode MoFEMErrorCode
MoFEM/PETSc error code.
Definition: Exceptions.hpp:56
implementation of Data Operators for Forces and Sources
Definition: Common.hpp:10
PetscErrorCode DMMoFEMTSSetMonitor(DM dm, TS ts, const std::string fe_name, boost::shared_ptr< MoFEM::FEMethod > method, boost::shared_ptr< MoFEM::BasicMethod > pre_only, boost::shared_ptr< MoFEM::BasicMethod > post_only)
Set Monitor To TS solver.
Definition: DMMMoFEM.cpp:1003
PetscErrorCode PetscOptionsGetInt(PetscOptions *, const char pre[], const char name[], PetscInt *ivalue, PetscBool *set)
SmartPetscObj< Vec > smartVectorDuplicate(SmartPetscObj< Vec > &vec)
Create duplicate vector of smart vector.
FormsIntegrators< BoundaryEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpBaseTimesScalar< 1 > OpBoundaryTimeScalarField
FormsIntegrators< BoundaryEleOp >::Assembly< PETSC >::BiLinearForm< G >::OpMass< 1, SPACE_DIM > OpBoundaryMass
[Only used with Hencky/nonlinear material]
Definition: plastic.cpp:71
constexpr double t
plate stiffness
Definition: plate.cpp:59
static constexpr int approx_order
Simple interface for fast problem set-up.
Definition: BcManager.hpp:23
Core (interface) class.
Definition: Core.hpp:82
static MoFEMErrorCode Initialize(int *argc, char ***args, const char file[], const char help[])
Initializes the MoFEM database PETSc, MOAB and MPI.
Definition: Core.cpp:72
static MoFEMErrorCode Finalize()
Checks for options to be called at the conclusion of the program.
Definition: Core.cpp:112
Deprecated interface functions.
Data on single entity (This is passed as argument to DataOperator::doWork)
Get field gradients at integration pts for scalar filed rank 0, i.e. vector field.
Get rate of scalar field at integration points.
Get value at integration points for scalar field.
Post post-proc data at points from hash maps.
Set indices on entities on finite element.
Set inverse jacobian to base functions.
Modify integration weights on face to take in account higher-order geometry.
PipelineManager interface.
boost::shared_ptr< FEMethod > & getDomainLhsFE()
boost::shared_ptr< FEMethod > & getBoundaryRhsFE()
Simple interface for fast problem set-up.
Definition: Simple.hpp:26
intrusive_ptr for managing petsc objects
PetscInt ts_step
time step number
MoFEMErrorCode getInterface(IFACE *&iface) const
Get interface refernce to pointer of interface.
Monitor solution.
static constexpr int saveEveryNthStep
SmartPetscObj< DM > dM
MoFEMErrorCode postProcess()
function is run at the end of loop
boost::shared_ptr< PostProcEle > postProc
MoFEMErrorCode operator()()
function is run for every finite element
MoFEMErrorCode preProcess()
function is run at the beginning of loop
WaveEquation(MoFEM::Interface &m_field)
MoFEM::Interface & mField
MoFEMErrorCode readMesh()
MoFEMErrorCode setupProblem()
MoFEMErrorCode boundaryCondition()
MoFEMErrorCode initialCondition()
boost::shared_ptr< std::vector< unsigned char > > boundaryMarker
MoFEMErrorCode assembleSystem()
MoFEMErrorCode solveSystem()
MoFEMErrorCode outputResults()
MoFEMErrorCode runProgram()
MoFEMErrorCode setIntegrationRules()
FormsIntegrators< BoundaryEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpBaseTimesScalar< 1 > OpBoundaryTimeScalarField
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::BiLinearForm< GAUSS >::OpMass< 1, 1 > OpDomainMass
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpBaseTimesScalar< 1 > OpDomainTimesScalarField
constexpr double wave_speed2
FormsIntegrators< BoundaryEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpSource< 1, 1 > OpBoundarySource
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::LinearForm< GAUSS >::OpGradTimesTensor< 1, 1, SPACE_DIM > OpDomainGradTimesVec
FormsIntegrators< DomainEleOp >::Assembly< PETSC >::BiLinearForm< GAUSS >::OpGradGrad< 1, 1, SPACE_DIM > OpDomainGradGrad
FormsIntegrators< BoundaryEleOp >::Assembly< PETSC >::BiLinearForm< GAUSS >::OpMass< 1, 1 > OpBoundaryMass